Caramelize the onions: In a medium skillet, melt 2 tablespoons of butter over medium heat. Add onion and cook, stirring occasionally, until soft and golden brown, about 25 to 30 minutes. Remove from heat and set aside.
Make the patties: In a large bowl, combine ground beef, Worcestershire sauce, salt, and pepper. Shape mixture into 4 thin, oval patties (about 6 inches by 4 inches each).
Cook the patties: Melt 1 tablespoon of butter in a large cast iron skillet over medium-high heat. Cook 2 patties until well browned on both sides and cooked through, about 2 minutes per side. Transfer to a plate. Repeat with another tablespoon of butter and the remaining 2 patties. Wipe skillet clean.
Make the sauce: In a small bowl, stir together the mayonnaise, ketchup, and relish. Set aside.
Assemble the sandwiches: Layer one slice of bread with 1 tablespoon of sauce, a slice of cheese, a beef patty, a couple tablespoons of caramelized onions, and another tablespoon of sauce. Top with another slice of bread. Repeat with remaining ingredients to create 4 sandwiches.
Griddle the sandwiches: Melt 1 tablespoon of butter in the large skillet over medium heat. Cook 2 sandwiches until golden brown and cheese is melted, about 3 to 4 minutes per side. Transfer to a baking sheet and keep warm in a 200°F oven. Repeat with remaining butter and sandwiches. Serve immediately.