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