Microwave 2 minced onions, 6 minced garlic, 2 Tablespoons ginger, and 1 Tablespoon oil in bowl, stirring occasionally; until onions are softened, about 5 minutes; transfer to slow cooker.
Stir 4 cups broth, 1 can coconut milk, grated lemon, 3 sliced carrots, 1 Tablespoon fish sauce, and 10 tied cilantro stems into slow cooker. Season chicken with salt and pepper and nestle into slow cooker. Cover and cook until chicken is tender, 4 to 6 hours on low.
Transfer chicken to cutting board, let cool slightly, then shred into bite-sized pieces using two forks to pull the chicken apart. Discard cilantro stems. Do not return chicken to soup just yet.
Stir in 8 ounces mushrooms, cover, and cook on high for 15 minutes. Microwave remaining 1 can coconut milk in bowl until hot, about 3 minutes, then whisk in remaining 2 Tablespoons fish sauce, 3 Tablespoons lime juice, 1 Tablespoon sugar and 2 teaspoons curry paste to dissolve.
Stir hot coconut milk mixture and shredded chicken into soup and let sit until heated through, about 5 minutes. Season with salt and pepper to taste and serve with garnishes.