Place onions in a colander and salt. Let sit for 30 minutes. Using a clean dish towel or paper towels, squeeze out as much liquid from onions as possible.
Divide the onions into fourths on a rimmed baking sheet. Divide ground beeg into fourths. Place the beef on top of the onions and, using your palm, flatten the beef, pressing it firmly into the onions. Season with salt and pepper.
Melt the butter and oil in a large skillet over medium heat. Transfer the patties to the skillet with the onions on the bottom. Cook for 6-8 minutes. Onions should begin to get crispy and turn a deep golden brown. Flip the burgers and increase the heat to high. Place the bottom half of the bun on top of the onions, and then the top half of the bun on top of the bottom half. Let the buns steam while the burgers cook an additional 2 minutes.
Remove the buns and set aside. Flip the burger so the onions again are on the bottom. Add 1 or 2 slices of cheese to each burger and allow to melt. Place the burger on the bottom half of the bun, add condiments and top half of bun. Serve immediately.
Adapted from A Taste of Home Cooking