Preheat a panini-style grill or George Forman style grill.
Heat the oil in a nonstick pan on medium heat. Saute the onions until they become soft and golden brown. Once golden brown, add in the mushrooms and sauté until soft. Remove from the pan and set aside.
In a bowl mix the mayo, garlic powder, salt and pepper. Taste and adjust the seasoning as needed. Spread the aioli on each slice of bread.
Place a slice of cheese on each slice of bread, layer with roast beef, mushrooms, onions, remaining cheeses, and the other slice of bread on top.
Cook the sandwich on the panini grill until golden brown and the cheese has fully melted. (alternatively cook the sandwich in a lightly greased nonstick pan on medium heat until the same results above have been achieved).