n a medium skillet, heat the oil over medium. Add the onions and bay leaf; season with salt and pepper. Cook, stirring often, until the onions are light brown and very soft, 20 to 25 minutes. Add the wine; stir until it evaporates, about 5 minutes. Discard the bay leaf. Heat a cast-iron skillet or griddle pan over medium-high.
Meanwhile, using a spice mill or clean coffee grinder, grind the mushrooms into a fine powder. In a medium bowl, combine the beef and mushrooms; season.
On a cutting board, combine the parsley, horseradish, lemon zest and garlic. Finely chop, adding extra horseradish if you like more heat. Mix the gremolata into the beef. Form into 4 patties, thinner in the centers for even cooking.
In a small dish, melt the butter in the microwave.
Cook the patties, turning occasionally and basting with the melted butter as they cook, until browned, about 8 minutes. During the last minute or so of cooking, slice or shred the Fontina (or slice or scoop the Taleggio) onto the patties. Tent the pan loosely with foil to melt the cheese.
Serve the burgers on lettuce-lined buns topped with the caramelized onions.