Thai Coconut Chicken Soup (Tom Kha Gai) With Mushrooms

Ingredients

• 20 ounces chicken stock
• 2-inch piece of galangal, peeled, half cut into slices, half cut into matchsticks
• 2 stalks lemongrass, ends trimmed, tough outer leaves removed, cut into 1-inch pieces
• 4 kaffir lime leaves, crumpled by hand
• 1 (13.5 ounce) can coconut milk
• 1 pound boneless, skinless chicken breasts, sliced thinly against the grain
• 1/2 pound mushrooms (straw mushrooms if possible, but otherwise cremini or button), ends removed, sliced
• 1 tablespoon fish sauce, plus more for serving
• 1 tablespoon juice from 1 lime, plus more for serving
• 4 hot red chilis (such as thai bird) ends trimmed, smashed with a knife
• handful fresh cilantro leaves and tender stems