French Onion Open-Faced Burgers

Ingredients

• 1 pound Ground Beef (80% lean or leaner)
• Salt and pepper, to taste
• 1/4 cup butter, divided
• 4 brioche sandwich buns, bottoms removed and toasted, tops cut into 1/2-inch pieces
• 3 sprigs fresh thyme
• 2 cloves garlic, crushed
• 3/4 cup shredded Gruyere cheese (about 12 ounces)
• 2 tablespoons butter
• 1 small Spanish onion, thinly sliced
• 1 cup water
• 1/2 cup sherry or vegetable broth
• 1 tablespoon Worcestershire sauce
• 1 teaspoon low-sodium beef base
• 1 teaspoon salt
• 1/2 teaspoon ground black pepper
• 3 tablespoons butter
• 1/2 cup shallots
• 2 tablespoons brandy
• 1/2 cup dry red wine
• 1/2 cup demi-glace
• 1 tablespoon Dijon-style mustard
• 1 tablespoon whole green peppercorns or peppercorn medley
• 1-1/2 teaspoons ground black pepper
• 1 teaspoon Worcestershire sauce
• 1/2 cup heavy cream