Chicken Burgers with Bourbon Barbecue Sauce and Mushrooms

Ingredients

• 1 Tbs. olive oil
• 1/4 cup (1 1/2 oz./45 g) finely chopped yellow onion
• 2 garlic cloves, minced
• 1/2 cup (4 fl. oz./125 ml) bourbon
• 1 cup (8 oz./250 g) ketchup
• 2 Tbs. Dijon mustard
• 2 Tbs. balsamic vinegar
• 1 1/2 Tbs. molasses
• Pinch of cayenne pepper
• 1/2 tsp. kosher salt
• 1/2 tsp. freshly ground black pepper
• 2 Tbs. olive oil
• 1 Tbs. unsalted butter
• 1/2 lb. (250 g) white or brown mushrooms, sliced 1/4 inch (6 mm) thick
• 1/2 tsp. chopped fresh thyme
• 3/4 tsp. kosher salt
• 1/4 tsp. freshly ground pepper
• <br> 1 1/2 lb. (750 g) ground chicken, preferably dark meat
• Kosher salt
• freshly ground pepper
• 1/4 lb. (125 g) Monterey jack cheese, sliced
• 4 seeded buns, split
• 4 romaine lettuce leaves