Salt the onions: Toss the thinly sliced onions with 1 teaspoon of salt in a bowl. Transfer to a colander and let sit for 30 minutes to draw out moisture. Afterward, wrap the onions in a clean dish towel and squeeze out as much liquid as possible.
Form the patties: Divide the drained onions into 4 equal mounds on a baking sheet or large cutting board. Form the ground beef into 4 equal balls and place each one on top of an onion mound. Press down firmly to flatten the beef into a 4-inch patty, pressing the onions into the meat. Season the exposed side with salt and pepper.
Sear onion-side down: In a 12-inch skillet, heat the butter and vegetable oil over medium heat. Carefully transfer each patty onion-side down into the skillet. Cook for 6–8 minutes without moving, allowing the onions to caramelize and crisp up against the pan.
Flip & finish: Increase heat to high and carefully flip each burger. Cook for 2 more minutes to brown the beef side. Add American cheese slices to the top during the last minute of cooking to melt.
Assemble: Spread mayo, mustard, and place pickles on the toasted buns. Add the burgers and serve immediately while hot and melty!
