Three premises are required for the ESS/Complementarian argument to succeed. “The head of Christ is God” must mean that the head of the Eternal Son of God, even prior to the incarnation, is God the Father. “Head” (kephale) must mean “authority over” or similar. There must be an intended analogy (or parallel) between the headship of God over Christ and the headship of husband over wife. If any one of these premises are found wanting, the edifice falls.

In 1 Corinthians 11 Paul writes, “But I want you to understand that the head of every man is Christ, the head of a wife is her husband, and the head of Christ is God” (v. 3). In this verse, the word “head” refers to one who is in a position of authority over the other, as this Greek word (kephale) uniformly does whenever it is used in ancient literature to say that one person is “head of” another person or group. So Paul is here referring to a relationship of authority between God the Father and God the Son, and he is making a parallel between that relationship in the Trinity and the relationship between the husband and wife in marriage. This is an important parallel because it shows that there can be equality and differences between persons at the same time. […]

Just as the Father and Son are equal in deity and are equal in all their attributes, but different in role, so husband and wife are equal in personhood and value, but are different in the roles that God has given them. Just as God the Son is eternally subject to the authority of God the Father, so God has planned that wives would be subject to the authority of their own husbands. (Wayne Grudem, Biblical Foundations of Manhood and Womanhood, pp. 48-49)

1 Corinthians 11:3 is a/the linchpin passage in the Eternal Subordination of the Son (ESS)-meets-Complementarianism argument. Denny Burk, the current President of the Council on Biblical Manhood and Womanhood, has said as much himself (see, e.g., HERE).  Three premises are required for the ESS/Complementarian argument to succeed.

1. “The head of Christ is God” must mean that the head of the Eternal Son of God, even prior to the incarnation, is God the Father.
2. “Head” (kephale) must mean “authority over” or similar.
3. There must be and intended analogy (or parallel) between the headship of God over Christ and the headship of husband over wife.

If any one of these premises are found wanting, the edifice falls. And though I doubt very much that an analogy or parallel is found in this passage or that Grudem’s understanding of kephale is correct, much more important to me is the claim that “Christ” in the passage is intended to refer to the Son of God according to His eternal divine nature, thereby eternally subordinating the Second Person of the Trinity in “role” or otherwise. If, alternatively, “Christ” in 1 Corinthians 11:3 refers to the mediatorial condescension of the Son of God in history—that is, the Son in His flesh—then not only is there no whiff of ESS in this passage, but there is also no longer any way to say that this passage gives a metaphysical basis for claiming there can be full equality of nature and attributes, yet also eternal subordination. If “Christ” means “the Son of God in His flesh”, then the passage actually shows that Christ is in fact less than the Father (John 14:28) due to His union with human nature; viz., He is subordinate by virtue of bearing a different nature, not according to the nature He shares with the Father in eternity.

So, in what follows, I have gone back through the history of interpretation of 1 Corinthians 11:3 to see what exactly is the consistent historical witness of the Church catholic with regard to the meaning of “Christ” in the passage. Though I do not personally have access to everything ever written on the subject, I think a pretty wide representation can be found below.  Literally every author I consulted prior to the middle of 20^thCentury has had much the same interpretation of this passage.  I welcome all to look for themselves and let me know what you find.  Indeed, taste and see that “Christ” in 1 Corinthians 11:3 is the Son united to the economy of His flesh.

(Note: I have provided links anywhere I was able.)

Ambrose (340-397)

Let God, then, be the Head of Christ, with regard to the conditions of Manhood. Observe that the Scripture says not that the Father is the Head of Christ; but that God is the Head of Christ, because the Godhead, as the creating power, is the Head of the being created. And well said [the Apostle] “the Head of Christ is God;” to bring before our thoughts both the Godhead of Christ and His flesh, implying, that is to say, the Incarnation in the mention of the name of Christ, and, in that of the name of God, oneness of Godhead and grandeur of sovereignty.

But the saying, that in respect of the Incarnation God is the Head of Christ, leads on to the principle that Christ, as Incarnate, is the Head of man, as the Apostle has clearly expressed in another passage, where he says: “Since man is the head of woman, even as Christ is the Head of the Church;” whilst in the words following he has added: “Who gave Himself for her.” After His Incarnation, then, is Christ the head of man, for His self-surrender issued from His Incarnation.

The Head of Christ, then, is God, in so far as His form of a servant, that is, of man, not of God, is considered. But it is nothing against the Son of God, if, in accordance with the reality of His flesh, He is like unto men, whilst in regard of His Godhead He is one with the Father, for by this account of Him we do not take aught from His sovereignty, but attribute compassion to Him. (On the Christian Faith, Book 4.31-33)

John Chrysostom (349-407)

“But the head of the woman is the man; and the head of Christ is God.” Here the heretics rush upon us with a certain declaration of inferiority, which out of these words they contrive against the Son. But they stumble against themselves. For if “the man be the head of the woman,” and the head be of the same substance with the body, and “the head of Christ is God,” the Son is of the same substance with the Father. “Nay,” say they, “it is not His being of another substance which we intend to show from hence, but that He is under subjection.” What then are we to say to this? In the first place, when anything lowly is said of him conjoined as He is with the Flesh, there is no disparagement of the Godhead in what is said, the Economy admitting the expression… (Homilies, on 1 Corinthians 11:3)

Augustine of Hippo (354-450)

And again, “The head of the woman is the man, the head of the man is Christ, and the head of Christ is God.” But again, if God is only all three together, how can God be the head of Christ, that is, the Trinity the head of Christ, since Christ is in the Trinity in order that it may be the Trinity? Is that which is the Father with the Son, the head of that which is the Son alone? For the Father with the Son is God, but the Son alone is Christ: especially since it is the Word already made flesh that speaks; and according to this His humiliation also, the Father is greater than He, as He says, “for my Father is greater than I;” so that the very being of God, which is one to Him with the Father, is itself the head of the man who is mediator, which He is alone. (On the Holy Trinity, 9.10)

Thomas Aquinas (1225-1274)

God is the Head of Christ, as man, according to 1 Cor. 11:3, “The Head of Christ is God.” (Summa Theologica, III, Q. 8, Art. 1, Obj. 2)

John Calvin (1509-1564)

Let us, for the present, take notice of those four gradations which he points out. God, then, occupies the first place: Christ holds the second place. How so? Inasmuch as he has in our flesh made himself subject to the Father, for, apart from this, being of one essence with the Father, he is his equal. Let us, therefore, bear it in mind, that this is spoken of Christ as mediator. He is, I say, inferior to the Father, inasmuch as he assumed our nature, that he might be the first-born among many brethren. (Commentary on 1 Corinthians 11:3)

Zacharias Ursinus (1534-1583)

Objection 8: The Son has a head and is less than the Father. Therefore he is not one and the same essence with the Father.

Answer: The Son has a head in respect to his human nature, and his office as mediator. These things, however, do not detract any thing from his Divinity. (Commentary on the Heidelberg Catechism, p. 374)

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