Heat oven to 350 °F [180ºC]. Grease a mid-sized baking pan.
2. Add ingredients
In a saucepan heat the butter over low heat. Add the flour and stir vigorously until the two are well mixed. Add the milk and stir until it reaches a uniform consistency. Add a pinch of salt and keep stirring until it becomes a thick cream. Add the mozzarella cheese and mix. Taste and add more salt to taste if needed. Set aside.
3. Bake
Lay a layer of eggplants on the baking pan and spread some of the sauce that you had prepared previously. Keep adding layers of eggplants, "glueing" each layer with the sauce. Cover with a final layer of eggplant. Cover the baking pan with tinfoil and bake for 40 minutes.
4. Finishing
Uncover, sprinkle with the parmesan cheese and bake until the top is light golden.
Notes
For a vegan version use cheese substitute of your choice (I have made it with rice "mozzarella".) The consistency and flavor will be different, but still delicious.The main point of using Parmesan is to obtain a crispy crust that contrasts with the creaminess of this dish. You can obtain a similar result by using breadcrumbs lightly soaked in olive oil.