Peel eggplant and cut into four pieces. Boil in a small amount of salted hot water for 10 minutes. Drain and squeeze out excess water. Remove the largest seeds.
In a bowl, combine eggplant pulp with eggs, a heaping tablespoon of grated cheese and enough breadcrumbs to thicken the mixture. Melt butter in a saucepan, but do not brown, and pour over the mixture.
Dust with nutmeg and combine well. Butter an ovenproof baking dish and sprinkle bottom and sides with remaining breadcrumbs. Pour mixture into the dish and bake in a preheated 325°F oven for 20 minutes. Serve very hot.