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Stir together mayonnaise, ketchup, relish, sugar, and vinegar in a small bowl for the sauce.
Heat butter and 1 tablespoon oil in a nonstick skillet over medium heat. Add onions and season with ¼ teaspoon salt. Cook, stirring occasionally, until onions are tender and golden brown, about 20 minutes. When skillet starts to become dry, add 1 tablespoon water. Continue cooking, stirring often, adding more water as needed, 1 tablespoon at a time, until onions are very soft, deeply browned, and jammy, 10 to 15 minutes more. You may not need to use all the water. Remove from heat.
Shape beef into 8 patties, about ⅜-inch-thick and 4 inches in diameter. Sprinkle with pepper and remaining ½ teaspoon salt.
Heat 1 teaspoon oil in a large cast-iron skillet over medium-high heat. Working in batches as needed, add patties to skillet. Spread 2 teaspoons mustard over top of each patty. Cook until browned, about 2 minutes per side. Top patties with cheese and onions. Cook, covered, about 30 seconds, until cheese is melted. An instant-read thermometer inserted into the center should read at least 160 degrees F (70 degrees C). Layer patties to create 4 stacks.
Spread sauce onto bun bottoms. Fill each bun with 4 pickles, a tomato slice, lettuce, and 1 patty stack.