Place the chicken pieces, sliced onion, carrot rounds, and red bell pepper strips into the slow cooker.
In a medium bowl, whisk together the coconut milk, red curry paste, fish sauce, brown sugar, minced garlic, and minced ginger until smooth.
Pour the coconut mixture over the chicken and vegetables in the slow cooker. Stir gently to coat everything evenly.
Cover and cook on low for 5 hours.
Add the broccoli florets and lime juice. Stir, then cover and cook for 1 more hour until the broccoli is tender but still bright.
Serve the curry over cooked white rice and sprinkle with fresh cilantro.