01 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions and cook, stirring frequently, for 15 to 20 minutes until soft and golden brown. Sprinkle sugar and salt halfway through cooking to enhance caramelization. Set aside.
02 - In a mixing bowl, combine ground beef, salt, black pepper, Worcestershire sauce, and Dijon mustard. Mix gently until just combined. Divide into eight equal portions and shape each into a patty slightly larger than the slider buns as they will shrink when cooking.
03 - Heat a grill pan or skillet over medium-high heat. Cook patties for 2 to 3 minutes per side until desired doneness. In the final minute, place a cheese slice on each patty and cover the pan to melt the cheese.
04 - Lightly toast slider buns, cut side down, in a dry skillet or on the grill until golden brown.
05 - Optional: Spread mayonnaise and mustard on the bottom half of each bun. Add a lettuce leaf or pickle slice if desired. Place the cheesy beef patty atop, spoon caramelized onions over the cheese, then cover with the top bun.
06 - Serve sliders immediately while warm.