Blend the mayonnaise, ketchup, Dijon, pickle juice, paprika, and garlic powder in a small bowl until perfectly smooth; refrigerate until needed.

Warm the olive oil in a large skillet over medium-low heat, add the onions, and cook slowly, stirring occasionally, until they turn a deep golden brown—about 30 minutes. If they begin to scorch, lower the heat and moisten with a splash of water. Transfer the onions to a dish and set aside.

Combine the ground beef with the salt, pepper, and Worcestershire sauce, then shape the mixture into four thin oblong patties that mimic the shape of the bread.

Increase the skillet to medium heat and sear the patties until browned on both sides. Remove to a plate and wipe excess grease from the pan.

Butter one side of each slice of rye. Build each sandwich in this order: bread (buttered side down), one slice Swiss, a beef patty, a generous layer of caramelized onions, a spoonful of secret sauce, one slice cheddar, and the second slice of bread (buttered side up).

Return the skillet to medium heat and grill the sandwiches until the underside is crisp and golden, 1–2 minutes; flip and toast the second side to the same rich color, allowing the cheese to melt through.

Halve each patty melt and serve hot, offering any remaining sauce on the side.