Stuffed Onion Soup Burgers

Ingredients

• 2 tbsp. butter -
• 1 large onion, thinly sliced -
• Kosher salt -
• Freshly ground black pepper -
• 1 tbsp. fresh thyme leaves -
• 2 tbsp. dry white wine -
• 1 lb. ground beef -
• 1 packet French onion soup mix (such as Lipton's) -
• 1 tbsp. Worcestershire sauce -
• 2 cloves garlic, minced -
• 1 large egg -
• 2 c. shredded Gruyère -
• 1 tbsp. extra-virgin olive oil -
• 4 hamburger buns -
• 4 pieces lettuce -
• 4 slices tomato -