Prepare the filling. Place the eggplant, zucchini, onion and carrot in a frying pan with heated vegetable oil. Fry until golden brown. Add the mushroom stems and garlic to the vegetables. Simmer for about 5 more minutes. Cool the filling and combine it with 2/3 of the cheese and parsley. Add salt and pepper. Mix.
Bake the mushrooms. Fill the mushroom caps with the prepared filling. Sprinkle the remaining cheese on top. Transfer to a baking dish and place in the oven, preheated to 190 degrees, for 25 minutes.