Step 1: Caramelize the onions. Heat olive oil and butter in a skillet over medium heat. Add the sliced onions and cook slowly, stirring often, for about 20 minutes until golden and caramelized. Reduce heat if they start to brown too quickly. Set aside.

Step 2: Prepare the burger patties. In a mixing bowl, combine ground beef, Worcestershire sauce, garlic powder, onion powder, salt, and pepper. Mix gently with your hands or a fork just until combined. Divide the mixture into 6 equal portions and form into patties slightly larger than your buns.

Step 3: Cook the burgers. Heat a large skillet or grill pan over medium-high heat. Cook the patties for about 4–5 minutes per side for medium, or until they reach your preferred doneness. In the last minute of cooking, top each patty with a slice of Irish cheddar and cover the pan to melt the cheese.

Step 4: Toast the buns. While the burgers cook, toast your buns either in the oven or on a skillet, cut side down, until golden and lightly crisped.

Step 5: Make the sauce. In a small bowl, mix together mayonnaise, Dijon mustard, and horseradish. Adjust to taste. This creamy, tangy sauce adds depth and ties the flavors together beautifully.

Step 6: Assemble the burgers. Spread sauce on the bottom half of each toasted bun. Add a cheesy burger patty, then top with caramelized onions, arugula or cabbage, and tomato slices if desired. Finish with the top bun and serve immediately.