1. Heat the vegetable oil in a large skillet over medium-low heat. Add the sliced onions, cover and cook, stirring occasionally, until golden and soft, about 15 minutes. Uncover, increase the heat to medium high and continue to cook, stirring often, until caramelized, about 5 more minutes.
2. Mix the mayonnaise, ketchup, relish and vinegar in a bowl; set aside. Shape the beef into 8 patties, about 4 inches wide and 1/2 inch thick.
3. Heat a griddle or skillet over medium heat; lightly brush with melted butter. Toast the buns on the griddle, split-side down. Spread each toasted bun bottom with about 1 tablespoon of the mayonnaise mixture, then top with a few pickles, some lettuce, 1 or 2 slices tomato and another dollop of the mayonnaise mixture; set aside.
4. Season both sides of the patties with salt and pepper. Working in batches if necessary, put the patties on the griddle and cook 5 minutes. Spread about 1 1/2 teaspoons mustard on the uncooked side of each patty, then flip and top each with 1 slice cheese; continue cooking about 3 more minutes for medium doneness. Top 4 of the patties with caramelized onions, then cover with the remaining patties, cheese-side up.