Chicken, Mushroom, And Rice Soup

Ingredients

• 1 (3 1/2) pound whole chicken
• 1 large carrot, cut into 2-inch pieces
• 1 rib celery, cut into 2-inch pieces
• 1 medium red onion, roughly chopped
• 1 large bay leaf
• water to cover
• ¼ cup unsalted butter
• 1 pound mushrooms, thickly sliced
• 1 teaspoon kosher salt, or more to taste
• 1 cup diced red onion
• 4 cloves garlic, minced
• ½ teaspoon freshly ground black pepper
• 1 pinch cayenne pepper
• 3 sprigs fresh thyme
• 1 cup medium-grain white rice