In a small bowl, mix mayonnaise, ketchup, and cayenne. Stir together until totally combined. Set aside. (This is the spicy fry sauce.)

Melt butter in a large skillet. Add onions and cook over medium heat for five minutes, stirring frequently. Add mushrooms and toss around, then add wine (if using), Worcestershire, salt and pepper. Cook for several minute over medium heat, or until all liquid has evaporated. Transfer mushroom mixture to a separate bowl, scraping all contents from the skillet.

Mix ground beef, heavy cream, Worcestershire, and salt and pepper in a bowl. Use your hands to thoroughly mix ingredients. Form 1/4 to 1/3 cup of the meat mixture into patties, making an indentation with your thumb to keep the patties from plumping too much when they cook. In the same skillet as the one used to cook the mushrooms, melt 1 tablespoon butter over medium to medium-high heat. Add four patties at a time, indented side face up. Cook for 4 minutes, then flip. Spoon generous portion of mushroom mixture on each patty, then top with one or two squares of Swiss, depending on your preference. Place lid on skillet and cook for an additional 2 to 3 minutes, until burgers are cooked through and cheese is melted. Remove to a place and keep warm while you repeat with the rest of the meat mixture.

Toast halved buns under the oven broiler. Spread generously with spicy fry sauce. Place patties between buns and serve immediately with fried onion strings.