Place the chicken thighs in the bottom of the slow cooker. Add the sliced onion, red bell pepper, and carrots on top.
In a bowl, whisk together the coconut milk, red curry paste, minced garlic, grated ginger, fish sauce, and brown sugar until smooth.
Pour the coconut mixture over the chicken and vegetables. Cover and cook on low for 6 hours or on high for 3 to 4 hours.
About 30 minutes before serving, cook the jasmine rice according to package directions.
Stir the lime juice into the curry. Shred or slice the chicken, then mix it back into the sauce.
Spoon the curry over the cooked jasmine rice and top with torn basil leaves.