Place the chicken thighs in the slow cooker and add the sliced red bell pepper, minced garlic, and grated ginger.
In a small bowl, stir together the soy sauce, fish sauce, and brown sugar until the sugar dissolves. Pour the mixture over the chicken and peppers.
Cover and cook on low for 5 to 6 hours or on high for 3 hours until the chicken is tender and shreds easily.
Shred the chicken with two forks right in the slow cooker and stir to coat it with the sauce.
Stir in the fresh Thai basil leaves and let them wilt for 5 minutes before serving.
Spoon the chicken and sauce over bowls of cooked rice and finish with lime wedges and sliced green onions.