Combine the onion slices and 1 teaspoon of salt in a bowl and toss to combine. Transfer the onions to a colander and let sit for 30 minutes, tossing occasionally. Transfer the onions to a clean dish towel, gather the edges together and squeeze out as much liquid as you can from the onions. Do this over the sink or a large bowl since the onions will have quite a bit of liquid in them.

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, cast iron would be perfect. Using a large spatula, transfer 2 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. Repeat for the remaining burgers.

Add 1 or 2 slices of cheese to each burger and allow to melt. Dress the burgers with what you like, such as mayo, mustard, ketchup, pickles, lettuce, etc. Serve.