From First Corinthians by George T. Montague, SM, commenting on I Corinthians 11:23-25:
What follows is the first historical witness to the institution of the Eucharist, predating the Gospel of Mark by at least five to ten years. But Paul already affirms that it is well-anchored tradition even in the details, the first of which is that it was instituted on the night on which Jesus was handed over, which should alert the Corinthians to the gravity of the occasion they are commemorating. The translation “handed over” instead of “betrayed” has the advantage of ambiguity about it, since Jesus was “handed over” not only by Judas (Mark 14:10, 42, 44) but also by the chief priests (to Pilate; Mark 15:1), and by Pilate (to crucifixion; Mark 15:15). It can also refer to God’s handing over Jesus for the salvation of the world (Rom 8:32).
Paul’s version of the words of institution are more than Mark’s and Matthew’s, which lack that is for you, and less than Luke’s (in 22:19), which has “given for you.” This cup is the new covenant in my blood echoes “This is the blood of the covenant” in Exod 24:8. The parallelism with the sacrifice’s sealing the old covenant demands taking “that is for you” as an affirmation of the sacrificial nature of this body, which brings salvation. The separate consecration of the bread and the wine signifies the separation of the blood from the body of Christ in death (though in reality the whole Christ is present under both species: bread and wine). The death of Christ is treated as a sacrifice in the Synoptics (Matt 20:28; Mark 10:45) and repeatedly in St. Paul. Here, then, the body of Christ is identified with the body immolated on the cross (so likewise John 6:51).