In a small bowl, combine mayonnaise, mustard, dill pickle relish, and ketchup. Stir until well blended and creamy, then refrigerate for at least 5 minutes.

Melt 2 tablespoons of butter in a skillet over medium-low heat. Add thinly sliced onions and season with Worcestershire sauce and red wine vinegar. Cook slowly for about 20 minutes until caramelized.

Shape ground beef into two equal patties, season with salt, pepper, and Worcestershire sauce. Cook in a hot skillet for about 4 minutes on one side, then flip and cook for another 3-4 minutes.

Butter one side of each slice of rye bread. In the skillet, layer cheese, burger sauce, caramelized onions, and a cooked patty. Top with another slice of bread.

Grill the sandwich for about 3-4 minutes on each side until golden brown and crispy. Let it rest for a minute before slicing.