• In a saucepan, melt the 2 Tablespoons butter Add the 1 large onion and cook over medium heat, stirring, until it begins to brown, about 10 minutes. Add the 2 Tablespoons sugar, 1 clove garlic , 3 sprigs fresh thyme and 1 bay leaf and continue to cook until the onion is deep golden, about 20 minutes.
• Add the 4 cups beef broth , 1 Tablespoon Worcestershire sauce and 1 teaspoon red wine vinegar and simmer, 30 minutes.
• Meanwhile, make the burgers.
• Divide the 1½ pounds ground beef into 4 equal portions and shape into patties. Season with salt and pepper.
• Heat 1 Tablespoon vegetable oil in a large cast-iron skillet over high heat. Once hot, add the burgers and cook, flipping once, until golden brown on both sides and cooked through, about 8 minutes total.
• Top each burger with a slice of cheese and cover the skillet. Heat for about 1 minute, until the cheese melts.

To Assemble

• Remove the thyme sprigs and bay leaf from the au jus. 

  Place the burgers on the buns along with creamy horseradish and arugula. Using a slotted spoon, remove some of the onions and top the burgers with them.
• Serve the remaining au jus on the side for dipping.