Place the chicken thighs, coconut milk, chicken broth, red curry paste, carrots, red bell pepper, onion, garlic, ginger, soy sauce, and brown sugar into the slow cooker.
Stir gently to combine the curry paste with the liquids.
Cover and cook on low for 6 to 7 hours, until the chicken is tender enough to shred.
Remove the chicken, shred it with two forks, and return it to the slow cooker.
Stir in the lime juice, then ladle the soup into bowls and top with fresh cilantro.