Place the chicken thighs in the slow cooker. Add the coconut milk, red curry paste, sliced onion, red bell peppers, carrots, garlic, ginger, fish sauce, and brown sugar. Stir to coat the chicken and vegetables evenly.
Cover and cook on low for 6 hours or on high for 3 to 4 hours until the chicken is tender and shreds easily.
Stir in the broccoli florets during the final 30 minutes of cooking so they soften but keep some texture.
Shred the chicken with two forks directly in the slow cooker, then stir in the lime juice.
Serve over cooked jasmine rice and top with fresh cilantro and Thai basil.