To make the aioli: Place the egg yolks, garlic cloves and lemon juice in the bowl of a food processor. Start the food processor and allow it to run while the oil is slowly and steadily drizzled in. You should have a creamy mayonnaise-like texture. Fold in the mustard and set aside for at least 30 minutes. Aioli can be kept in a covered container in the refrigerator.

In a small saute pan, heat 1 T olive oil over medium heat. Add the onions and cook slowly until soft, golden brown, and caramelized. Repeat with the mushrooms.

Heat a grill to medium heat. Form the ground beef into a patty and season it. Sprinkle it with the worcestershire; poke several small holes with a fork to allow it to absorb into the patty. Cook the patty on the grill, flipping once, until the burger is cooked to desired doneness, about 7 minutes total.

Top each patty with a slice of cheese, then half the sautéed onions and mushrooms, then the other slice of cheese. Turn the grill heat to low and allow the cheese to melt.

While you are waiting for the cheese to melt, add the hamburger buns to the grill, cut side down, to toast.

Stack the burger between the buns, smearing 1 T aioli on the top bun. Serve immediately.