Whisk 3 tablespoons mayonnaise, 1 tablespoon ketchup, 2 teaspoons sweet pickle relish, and ¼ teaspoon pepper together in bowl; set aside for serving.

Divide 1½ pounds 90 percent lean ground beef into twelve 2‐ounce portions, then roll into balls. (Balls can be shaped up to 24 hours in advance.)

Adjust oven rack to lowest position and heat oven to 450 degrees. Combine 1 finely chopped onion, 1 teaspoon vegetable oil, ⅛ teaspoon salt, and pinch pepper on rimmed baking sheet, then spread into even layer. Roast until onions are softened, 10 to 12 minutes, stirring onions and redistributing into even layer at least 3 times during roasting.

Arrange onions into 12 piles evenly spaced across sheet. Sprinkle meat balls with remaining ¼ teaspoon salt and remaining ⅛ teaspoon pepper. Place balls on top of onion piles. Using bottom of greased drinking glass or meat pounder, firmly smash each ball until about 3½ inches in diameter. Return sheet to oven and cook until edges of patties are set and tops are still slightly pink, 7 to 8 minutes.

Stack 6 slices deli American cheese and cut into quarters (you will have 24 pieces). Remove sheet from oven. Top each patty with 2 pieces of cheese. Place bottoms of 12 (2½‐inch) slider buns or soft dinner rolls on top of cheese and stack bun top on top of bun bottoms. Return to oven and cook until buns are warmed through, about 2 minutes.

Remove sheet from oven and remove bun tops. Transfer burgers to platter, carefully flipping each as you go so bun bottoms are underneath burgers and onions are on top of burgers. Top with reserved sauce and bun tops. Serve.