Heat vegetable oil in a large skillet over medium-low heat. Add the onions and 1/2 teaspoon salt, cover and cook, stirring occasionally, until golden and soft, about 30 minutes. Lower the heat more if the onions look like they’re cooking too quickly. After the half hour, uncover, increase the heat to medium-high and continue to cook, stirring often, until caramelized, about 8 more minutes. Add 1/4 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. You can caramelize the onions up to three days beforehand, cover, and refrigerate; then reheat before using. Mix the mayonnaise, ketchup, relish and vinegar in a bowl; set aside. Shape the beef into four patties, about 5-inches wide. Heat a skillet over medium heat; lightly brush with vegetable oil. Toast the buns on the skillet, 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 skillet and cook 3 minutes. Spread the mustard on the uncooked side of each patty, then flip and cook for 2 1/2 minutes.
Top each patty with cheese and cook for only 30 more seconds for medium doneness.
Place the patties on the dressed bun halves, add caramelized onions, and the other bun halves.