Wash and dry the eggplants, then slice them into moderately thick rounds. Place them in a colander with coarse salt to draw out bitterness. Let them sit for 2 to 3 hours, as these are more bitter than usual eggplants.

Wipe the mushrooms clean with a damp cloth and slice them. Season with salt and pepper. Drain the mozzarella, pat dry with paper towels, and slice it. Melt a little butter and brush both sides of the bread slices.

Pat the eggplant slices dry and grill them on a hot, oiled grill. Line a deep baking dish with the bread slices, covering the sides and bottom. Add a layer of eggplant and cover with tomato sauce. Top with chicken ham and sprinkle with shredded Emmental cheese. Add the grilled mushrooms and sprinkle with parsley. Peel and cut the avocado into pieces, drizzle with lime juice. Top the casserole with mozzarella and avocado. Bake in a preheated oven at 350°F (180°C) until golden brown.