Microwave onions, garlic, ginger, and oil in bowl, stirring occasionally; until onions are softened, about 5 minutes; transfer to slow cooker.
Stir broth, 1 can coconut milk, grated lemon, carrots, 1 tablespoon fish sauce, and 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 mushrooms, cover, and cook on high for 15 minutes. Microwave remaining can of coconut milt in bowl until hot, about 3 minutes, then whisk in remaining 2 tablespoons fish sauce, lime juice, sugar and 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.