In a large bowl, gently stir the ground beef, salt, pepper, and rosemary together. Divide into 6 even patties and set aside.
1½ pounds 85% lean ground beef, 1 teaspoon kosher salt, ½ teaspoon ground black pepper, 1 teaspoon minced fresh rosemary
In a cast iron skillet, melt the 3 tablespoons of butter for the mushrooms over medium-high heat. Add in the onion and cook until softened and golden, about 10-12 minutes. Add in the mushrooms and cook until caramelized, about 10 minutes.
3 tablespoons unsalted butter, 1 yellow onion, 8 ounces sliced mushrooms
Stir in the balsamic vinegar plus 2 tablespoons of water, scraping up the browned bits from the bottom of the pan. Season with salt and pepper to taste and set aside.
1 tablespoon balsamic vinegar, Salt and pepper
Wipe out the skillet and return it to medium-high heat. Spread each bun top and bottom with ½ tablespoon butter each. Place butter side down into the skillet and cook until golden and toasted, about 3-5 minutes. Remove from the pan and set aside.
6 brioche hamburger buns, 6 tablespoons unsalted butter
Place the patties into the skillet. Cook until the bottom side has formed a nice dark brown crust, flip and cover each patty with cheese. Cover and cook for 5-6 minutes until the cheese is melted.
6 slices Swiss cheese
To assemble the burgers, brush the bottom bun with mayonnaise, if desired. Place the patty on top, then mushrooms, arugula and the top bun. Enjoy!
Mayonnaise, A few handfuls of arugula