Juicy French Onion Burgers with Gruyère (Skillet or Grill)

Ingredients

• 2 large yellow onions, thinly sliced
• 1 tablespoon unsalted butter
• 1 tablespoon olive oil
• 1/2 teaspoon kosher salt
• 1/2 teaspoon sugar, optional
• 1/2 cup (120 ml) beef broth, divided
• 1 teaspoon fresh thyme leaves
• 1 teaspoon sherry vinegar or balsamic vinegar, optional
• 1 1/2 lb (680 g) 80/20 ground beef
• 1 tablespoon Worcestershire sauce
• 1 teaspoon garlic powder
• 1 teaspoon kosher salt
• 1/2 teaspoon black pepper
• 4 slices Gruyère cheese, about 4 oz (115 g)
• 4 brioche hamburger buns
• 2 tablespoons softened butter, for toasting buns
• 2 tablespoons Dijon mustard, optional
• Baby arugula or extra thyme, optional for serving