In a small bowl, combine mayonnaise, mustard, dill pickle relish, and ketchup. Mix well, cover, and refrigerate for later.

In a skillet over medium-low heat, add sliced onions, 2 tablespoons of butter, red wine vinegar, and Worcestershire sauce. Cook for about 20 minutes with the lid on, stirring occasionally. Once caramelized, transfer the onions to a plate.

While the onions are cooking, form two hamburger patties from the ground beef. Season both sides with salt and pepper. Drizzle a little Worcestershire sauce on the patties and rub mustard on both sides. Make sure the patties are slightly larger than the bread, as they will shrink while cooking.

Heat a cast iron skillet or regular skillet to medium-high heat. Place the burger patties in the skillet and cook for about 4 minutes. Flip the burgers and cook for another 3-4 minutes for a medium-well done burger.

Butter four slices of rye bread. Place two slices, butter side down, in the same skillet used for the onions. Spread burger sauce on the top side of all four slices of bread. On the two slices in the skillet, add a slice of cheddar cheese, followed by the caramelized onions, then the burger patty. Top with a slice of muenster cheese and finish with the other slice of rye bread, burger sauce side down.

Increase the skillet heat to medium-high. Cook until the bottom slice of rye is golden brown. Carefully flip the sandwich and grill the other side until golden and the cheeses are melted. Press down gently on the sandwich with a spatula for even grilling.