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Inspired by the Oklahoma onion burger, this weeknight recipe is totally irresistible. Round out the meal with fries or a salad.
Form the patties: Shape beef into 4 (1-inch-thick) patties. Use your thumb to make an indent in center of each patty. Season with salt and pepper. Heat oil in a large cast-iron skillet or griddle over medium-high.
Cook patties and onion: Place 2 patties in skillet. Place 1/3 cup onion next to each patty; cook for 2 minutes, leaving onion undisturbed for first minute and gently stirring during second minute.
Add cheese: Flip patties and place onion on top of patties; cook for 1 minute. Top onion with 1 slice cheese. Cook until cheese is melted and burger has reached just under desired degree of doneness, about 2 minutes for medium-rare and about 5 minutes for medium-well. Transfer patties to a plate. Repeat with remaining 2 patties, onion, and cheese. Remove skillet from heat to let cool slightly.
Assemble the patty melts: Place bottom buns, cut side down, on a clean work surface. Top with patties, then pickles. Place bun tops on patties, cut side up.
Toast the buns and serve: Return skillet to medium heat and add 1 tablespoon butter. Place 2 burgers in skillet; cook until golden brown, about 1 minute. Flip carefully and cook until golden brown on other side. Remove from skillet. Repeat with remaining 1 tablespoon butter and 2 burgers. Serve with sweet potato fries, if desired.