Clean the mushrooms, cut them into slices and fry them in about 100 ml hot olive oil in portions until golden brown. Remove from the pan and drain on paper towels, season with salt and pepper.
Peel the onions and garlic. Cut the onions into rings, finely chop the garlic, rinse the thyme and remove the leaves. Heat the remaining 50 ml of olive oil in the pan. Fry the onion rings in it for about 2 minutes, then add the garlic, vinegar and thyme and season with salt, pepper and a pinch of sugar.
Spread the mushroom slices evenly on plates and spread the warm onion vinaigrette over them.
Let cool and serve sprinkled with parmesan cheese.