01 - Melt butter with olive oil in a large skillet over medium heat. Add onions and salt. Cook, stirring frequently, for 15–20 minutes until deeply golden and soft. Stir in sugar during the last 5 minutes if using. Remove from heat and set aside.
02 - Combine ground beef, Worcestershire sauce, garlic powder, 1 teaspoon salt, and pepper in a bowl. Mix gently until just combined. Divide mixture into 8 equal balls and flatten into patties slightly larger than the buns to account for shrinkage.
03 - Preheat a grill, grill pan, or skillet over medium-high heat. Cook patties for approximately 2–3 minutes per side for medium doneness. During the final minute of cooking, place a slice of cheddar on each patty and cover to melt the cheese.
04 - Split the slider buns and toast them cut-side down on the grill or skillet for 1–2 minutes until lightly golden and crisp.
05 - Spread mayonnaise and mustard on the bottom buns if desired. Layer with arugula or lettuce, followed by the cheesy beef patty and a generous spoonful of caramelized onions. Top with the bun lids.
06 - Arrange the sliders on a platter and serve immediately while warm.