Preheat oven to 375˚F.
For the onions, in a skillet over medium heat, melt butter, add onions and season with salt and pepper, add a bay leaf. Cook onions, low and slow, to light caramel color, 20-25 minutes. Add 1 cup water, and cook over low heat until liquid is completely absorbed, 10 minutes more.
Meanwhile, for the mushrooms, preheat a second skillet over medium-high heat, add EVOO, 3 turns of the pan, and add mushrooms to brown, 10-12 minutes. Add thyme, garlic, and salt and pepper, and cook 5 minutes more. Add sherry or wine, let it absorb completely, turn off the heat, then add lemon juice.
Arrange bacon on slotted pan or wire-rack lined baking sheet and bake until crisp, 12-15 minutes. Mix together ingredients for horseradish sauce.
For the burgers, preheat a large cast-iron griddle or skillet over medium-high heat. Place beef in bowl. Using a handheld grater, grate onion over bowl so meat collects juices. Season with salt, pepper and rosemary. Add garlic, Balsamic and EVOO, mix into meat, and form 4 patties thinner at center and thicker at edges for even cooking. Cook patties 8 minutes, turning occasionally, melt 2 slices cheese on each patty.
Assemble burgers: bun bottom, horseradish sauce, bacon, sliced tomato, burger patty, mushrooms, onions, bun top.