Cut the chayotes in half, place in boiling salted water and cook until tender, approximately 15 min. Scoop out the pulp and reserve the shells.
Chop the pulp and drain. Mince the onion and fry slowly in butter until transparent. Add the pulp of the chayote and season to taste with salt and pepper. Break the two eggs over the mixture and cook over low heat for a few minutes, mixing well until done. Stuff the chayote shells and top with grated cheese. Place in a hot oven 400 degrees F. until melted and golden.