01 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions and salt. Cook, stirring frequently, for 15 to 20 minutes until onions are deeply golden and soft. Stir in sugar during the last 5 minutes if desired. Remove from heat and set aside.
02 - Combine ground beef, Worcestershire sauce, garlic powder, salt, and pepper in a medium bowl. Mix gently until just combined. Divide into 8 equal portions and shape each into a patty slightly larger than the buns.
03 - Heat a grill, grill pan, or skillet over medium-high heat. Cook patties for 2 to 3 minutes per side for medium doneness. Place a slice of cheddar on each patty in the last minute and cover to melt the cheese.
04 - Split slider buns and toast cut-side down on the grill or skillet for 1 to 2 minutes until lightly golden.
05 - Spread mayonnaise and/or Dijon mustard on the bottom buns if using. Layer with arugula or lettuce, followed by the cheddar-topped beef patty and a generous spoonful of caramelized onions. Cap with top bun and serve immediately.