Tom Kha Gai (Chicken Coconut Soup)

Name: Tom Kha Gai (Chicken Coconut Soup)
Cuisine: Thai
Ingredients: 1 1” piece ginger, peeled; 10 makrut (Thai) lime leaves or 1 Tbsp. lime zest and ¼ cup lime juice; 6 cups low-sodium chicken broth; 1½ lb. skinless, boneless chicken thighs, cut into 1” pieces; 8 oz. shiitake, oyster, or maitake mushrooms, stemmed, caps cut into bite-size pieces; 1 13.5-oz. can coconut milk; 2 Tbsp. fish sauce (such as nam pla or nuoc nam); 1 tsp. sugar; 2 stalks fresh lemongrass, tough outer layers removed; Chili oil, cilantro leaves with tender stems, and lime wedges (for serving)