Peel , cut and boil the eggplants with a little salt. Put them on a strainer with a weight to help loose all the water.
Peel , cut and boil the potatoes with a little salt. Use a potato ricer to mash them.
Preheat oven at 350 F.
In a small pan on medium heat put the olive oil with the butter ( reserve a tablespoon of butter ) until melted. Add the garlic and parsley an toss them for about 2 minutes or until the garlic gets golden.
Mix the eggplants with the mashed/riced potatoes , add the garlic/parsley/oil/butter mixtrue , add also the egg and parmsean cheese. Reserve a tbsp.of cheese.
Butter a 13 x 9 oven baking pan , pour the potaoes/ eggplant mixture , sprinkle with the reserved cheese and butter.
Put it on the oven for 30' or until cheese looks golden.
Serve warm as a main dish , or as a hearty side.
It's also delicious cold!