Heat the vegetable oil in a large skillet over medium-low heat. Add the onions and 3/4 teaspoon salt, cover and cook, stirring occasionally, until golden and soft, about 30 minutes. (If the onions brown too quickly, reduce the heat to low.) Uncover, increase the heat to medium high and continue to cook, stirring often, until caramelized, about 8 more minutes. Add 1/2 cup water to the skillet, scraping up any browned bits from the bottom of the pan. Simmer, stirring, until the water evaporates, about 2 more minutes; transfer to a bowl and set aside. (The onions can be made up to 3 days ahead; cover and refrigerate, then reheat before using.)
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.
Heat a griddle or skillet over medium heat; lightly brush with vegetable oil. 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. (Keep the griddle hot.)
Season both sides of the patties with salt and pepper. Working in batches if necessary, put the patties on the griddle and cook 3 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 2 more minutes for medium doneness. Top 4 of the patties with caramelized onions, then cover with the remaining patties, cheese-side up. Sandwich the double patties on the buns.