Place a non-stick skillet over medium-low heat. Add sliced onions and 2 tablespoons of water. Cover and cook for 10 minutes until softened.
Remove lid from onions, increase heat to medium, and pour in Guinness. Stir frequently until the liquid has evaporated and onions are dark brown and jam-like. Set aside.
Divide the lean ground beef into four equal 6-ounce portions. Form into patties approximately 3/4-inch thick, creating a slight indentation in the center of each to prevent bulging.
Season both sides of the patties uniformly with salt, black pepper, and garlic powder.
Heat a large cast-iron skillet or grill over medium-high heat and coat lightly with cooking spray.
Sear the patties for 4-5 minutes on the first side. Flip the patties and immediately brush with Worcestershire sauce.
Top each patty with 1 ounce of Irish cheddar. Cover the skillet with a lid for 1-2 minutes to melt the cheese while finishing the cook to an internal temperature of 160°F (71°C).
Lightly toast the protein buns on the grill or in a separate pan.
Assemble by spreading Dijon mustard on the bottom bun, followed by arugula, the cheeseburger patty, and a generous portion of Guinness onions.