In a small bowl, mix together mayonnaise, pickle relish, ketchup, and sugar. Stir until smooth and set aside.
Heat some vegetable oil in a pan over medium heat. Add finely diced onion and a pinch of salt. Stir occasionally and cook until golden and soft. Add water, a tablespoon at a time, if onions start to stick, until fully caramelized. Remove from heat.
In a large skillet, melt some butter over medium-high heat. Add ground beef, season with salt and pepper. Stir in Worcestershire sauce, beef broth, and yellow mustard. Cook, breaking up the beef, until browned. Divide the beef into portions, top with American cheese slices, and let the cheese melt. Remove from heat.
In a large bowl, layer shredded lettuce at the bottom. Add the cooked ground beef on top of the lettuce. Scatter dill pickle slices, chopped pepperoncinis, parsley, and the caramelized onions over the beef. Drizzle with the prepared burger sauce and toss everything together right before serving.
