Diner-Style Patty Melt

Season beef with salt and pepper; divide into 4 patties. Form each patty into a thin oval that is slightly larger than a slice of bread you’re using. Heat 1 Tbsp. oil in a large skillet or griddle, preferably cast iron, over medium-high. Cook onion slices in a single layer, being careful not to let rings separate, until softened and charred, about 4 minutes per side. Transfer to a plate and divide into individual rings. Add remaining 1 Tbsp. oil to skillet. Working in batches if needed, cook patties, pressing flat with a spatula, until browned but still pink in the center, about 1 1/2 minutes per side. Transfer to a plate. Wipe out skillet and reduce heat to medium-low. Top each slice of bread with 1 cheese slice, 1 patty, one-quarter of the onions, another slice of cheese, then a second slice of bread. Spread tops of each sandwich with 1 tsp. mayonnaise. Working in batches if needed, cook sandwiches, mayo side down, spreading the top of each sandwich with 1 tsp. mayonnaise, until golden brown, about 3 minutes. Flip and continue to cook until bread is golden brown and cheese is melted, about 3 minutes.