In a large skillet, melt 1 tbsp butter over medium-low heat. Add the sliced onions and a pinch of salt. Cook, stirring occasionally, for 10-15 minutes until softened and beginning to brown. Add 1 tsp brown sugar and continue to cook for another 5 minutes, until deeply caramelized.

Pour in the 1/4 cup Guinness stout. Bring to a simmer and cook until the liquid has almost completely evaporated and the onions are glossy and tender, about 5-7 minutes. Season with a pinch of salt and pepper. Remove from heat and set aside.

In a small bowl, whisk together the mayonnaise, Dijon mustard, prepared horseradish, chopped chives, lemon juice, salt, and pepper until well combined. Taste and adjust seasoning as needed. Cover and refrigerate until ready to use.

In a large mixing bowl, gently combine the ground beef with Worcestershire sauce, garlic powder, onion powder, 1 tsp salt, and 1/2 tsp black pepper. Avoid overmixing. Divide the mixture into 4 equal portions and form into 3/4-inch thick patties, slightly wider than your burger buns, as they will shrink during cooking. Press a shallow indentation in the center of each patty to prevent bulging.

Heat a large skillet or cast iron pan over medium-high heat. Add the burger patties and cook for 4-5 minutes per side for medium-rare, or longer to your desired doneness. During the last minute of cooking, place a slice of Irish cheddar on top of each patty, cover the pan briefly, and allow the cheese to melt.

While the burgers are cooking, lightly butter and toast the brioche buns in a separate pan or under a broiler until golden brown. Be careful not to burn them.

Spread a generous amount of Irish Pub Sauce on both halves of each toasted bun. On the bottom bun, layer a piece of green leaf lettuce, a slice of tomato, and a few pickle chips. Place a cheesy burger patty on top. Spoon a generous amount of the stout caramelized onions over the burger patty. Top with the other half of the bun.

Serve immediately with your favorite pub-style sides like fries or a crisp salad.