In a small saucepan over medium heat, pour the Guinness stout and allow it to reduce until it becomes a thick glaze. This should take about 10 minutes—set aside to cool.

In a large mixing bowl, combine the ground beef, cooled Guinness reduction, chopped onion, minced garlic, salt, and pepper. Mix gently with your hands to incorporate all ingredients evenly.

Shape the mixture into four equal patties, handling them as little as possible to maintain tenderness.

Preheat your grill or frying pan over medium-high heat. Cook each patty for 3-4 minutes on each side for medium rare, or longer depending on your preference.

During the last minute of cooking, place a slice of cheddar cheese on each patty, allowing it to melt.

Toast the burger buns lightly on the grill or in a toaster for added texture.

Assemble the burger by spreading whole grain mustard on the bottom bun, then adding a patty, followed by arugula, and finally, the top bun. Serve immediately.