Heat the butter and oil in a frying pan, then briskly sauté the mushrooms until they get a good colour. Season to taste. You need to cook them until they have thrown out all their moisture (and it has evaporated), otherwise you will end up with a very watery dish.
Add the thyme and garlic and cook for another minute.
Spoon the mushrooms into a small ovenproof dish and make four little holes into which you can drop the eggs. Break an egg into each hollow, spoon on the cream and scatter on the gruyère. It’s important that none of the eggs touch the side of the dish, otherwise they will cook faster than those in the middle.
Bake for 12 minutes, or until the eggs are set (but the yolks are still soft). Serve immediately with bread.