Just to add a little point of information to what Fr David said, while the priest may certainly chant the entire Eucharistic Prayer, there are clear instructions saying that the prayer is to be proclaimed by the priest alone and all musical instruments are to be silent:
GIRM: "32. The nature of the "presidential" texts demands that they be spoken in a loud and clear voice and that everyone listen with attention.44 Thus, while the priest is speaking these texts, there should be no other prayers or singing, and the organ or other musical instruments should be silent."
Redemptionis Sacramentum: "52. The proclamation of the Eucharistic Prayer, which by its very nature is the climax of the whole celebration, is proper to the Priest by virtue of his Ordination. It is therefore an abuse to proffer it in such a way that some parts of the Eucharistic Prayer are recited by a Deacon, a lay minister, or by an individual member of the faithful, or by all members of the faithful together. The Eucharistic Prayer, then, is to be recited by the Priest alone in full.
[53.] While the Priest proclaims the Eucharistic Prayer “there should be no other prayers or singing, and the organ or other musical instruments should be silent”, except for the people’s acclamations that have been duly approved..."
I only know of one priest in my diocese who regularly chants the Eucharistic Prayer in its entirety, but I think it adds enormously to the beauty of the prayer. If nothing else, chanting of the E.P. could slow down some priests who would otherwise have a tendency to rush through it!