In a medium heat-proof bowl, microwave onions, garlic, ginger, and oil. Cook for 5 minutes, stirring occasionally, until the onions are softened.
In the slow cooker add the broth, 1 can of coconut milk, lemon grass, carrots, 1 tablespoon fish sauce, and cilantro stems. Give it a quick stir, then add the chicken to the pot, nestling it into place. Cover and cook on low until the chicken is tender, about 4-6 hours.
Transfer chicken to cutting board, and let it cool slightly before shredding with 2 forks. Let the soup sit for 5 minutes, then remove the fat from the surface. Remove lemon grass and cilantro and discard.
Stir in the mushrooms, cover and cook on high until mushrooms are tender, 10-15 minutes. Microwave the remaining can of coconut milk in a small bowl until warm, about 1-2 minutes. Add it to the pot along with the remaining 2 tablespoons of fish sauce, lime juice, sugar, and curry paste.
Add the shredded chicken and let it heat through, about 5 minutes. Season with salt and pepper and serve with garnishes.