Preheat the oven to 180°C/350 °F.
Slice the eggplant into pieces lengthwise, about 1/2 inch thick. In a bowl, mix together the eggs, flour and a pinch of salt—if it is too thick for dipping, add the milk.
Pour the oil into a frying pan, estimating an amount that will cover half the eggplant pieces, and heat over a medium–high flame until little bubbles begin to form.
Dip the pieces of eggplant in the batter and fry them in the oil until they are crisp, about 2 minutes on each side. Keep the oil very hot while you are frying them. Remove them from the oil with a slotted spoon or spatula and drain on absorbent paper.
If using fresh tomatoes, crush them with a fork in a small bowl.
Layer 1/3 of the eggplant pieces in the bottom of a medium-sized baking pan. Cover the eggplant with half the mozzarella, 1/3 of the crushed tomatoes, salt, black pepper, and half of the grated parmesan. Repeat the process until you have 3 layers of eggplant and 2 of filling, and over the last layer of eggplant put the remainder of tomato, a little olive oil, and some pepper.
Bake for 20 minutes, and serve with additional grated parmesan.