Heat a 12-inch heavy-bottomed pan over high heat until very hot. Add the oil and then the mushrooms and cook, stirring, until golden brown, about 5 minutes. Add the lemon juice and wine and cook, 2 to 3 minutes. Stir in the butter and garlic and cook, stirring, until the butter is absorbed. Cool, then chill completely. Mince the mixture finely and store in an airtight container until ready to use.
For the burger:
Combine the beef, wine, Worcestershire sauce, garlic powder, Sriracha, salt and pepper to taste and 1 teaspoon minced mushrooms in a medium bowl and mix well. (Save the rest of the mushrooms for another use.) Form into one thick patty.
Lay 2 pieces of Texas toast on a work surface. Arrange 2 slices of American cheese on each of the 2 pieces (using a total of 4 slices), then layer 2 slices of Cheddar over each (using a total of 4 slices) and 2 slices of Swiss cheese over each (using a total of 4 slices). Top each with the remaining 2 pieces of Texas toast, and butter the outside of each sandwich.
Heat a large griddle over high heat. Cook the patty to desired doneness (about 4 minutes per side for medium-rare) and grill the cheese sandwiches until golden brown, about 3 minutes on each side. Lay bacon flat on the griddle and cook, 5 to 7 minutes. Flip and cook an additional 3 to 5 minutes, then reserve on paper towels to drain. Slather 1 side of the hot grilled cheese sandwiches with mayonnaise and mustard. Add the remaining 2 slices of American cheese, 2 slices of Cheddar, and 2 slices of Swiss cheese to the patty, and place the patty on one of the grilled cheese sandwiches. Top with the bacon, tomato, lettuce and onion. Top with the second grilled cheese. Serve hot with a pickle spear and chips.