Divide the onions into 4 separate mounds on a rimmed baking sheet. Form the beef into 4 lightly packed balls. Place the beef balls on top of the onion mounds and flatten the beef firmly (use your palm) so the onion adheres to the beef. The patties should measure 4 inches in diameter. Season the beef generously with salt and pepper.
Melt the butter with the oil in a 12-inch skillet over medium heat. Using a large spatula, transfer the patties to the skillet, onion side down. Cook for 6-8 minutes, until the onions turn a deep golden brown and begin to crisp around the edges. Flip the burgers, then increase the heat to high and cook until well browned on the second side, about 2 minutes. Add 1 or 2 slices of cheese to each burger and allow to melt. Add mayo, mustard and pickles to the top buns, then place each burger on a bottom buns. Serve immediately.
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