Heat the vegetable oil in a large skillet over medium-low heat. Add the onions and season with 3/4 teaspoon salt; cover and cook, stirring occasionally, until golden and soft, about 30 minutes. (Reduce the heat to low if the onions are browning too quickly.) Uncover, increase the heat to medium high and continue to cook, stirring often, until caramelized, about 8 more minutes. Add 1/2 cup water, scraping up any browned bits from the bottom of the pan. Simmer, stirring, until the water evaporates, about 2 minutes; transfer the onions to a bowl and set aside.
Mix the mayonnaise, ketchup, relish and vinegar in a small bowl; set aside. Form the beef into 8 patties, about 4 inches wide and 1/2 inch thick.
Heat a griddle or large skillet over medium heat; lightly brush with vegetable oil. Toast the buns cut-side down on the griddle. Spread each bun bottom with about 1 tablespoon of the mayonnaise mixture, then top with a few pickles, 1 or 2 tomato slices, some lettuce 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, transfer the patties to the griddle and cook 3 minutes. Spread about 1 1/2teaspoons mustard on top of each patty, then flip and cover with 1 cheese slice; cook 2 more minutes for medium. Top 4 of the patties with the caramelized onions, then cover with the remaining patties cheese-side up. Transfer the double patties to the bun bottoms; cover with the bun tops.