Cut the chicken breasts into 1-inch pieces and place them in the greased 6-quart slow cooker insert.
Add the coconut milk, chopped ginger, jalapeño, garlic, basil, curry powder, cumin, turmeric, salt, pepper, cinnamon, lime juice, and sugar to a food processor or blender.
Process or blend until the jalapeño and basil are finely chopped and the mixture is fairly smooth.
Pour the blended mixture over the chicken in the slow cooker, cover, and cook on LOW for 3 hours 45 minutes.
Whisk the cornstarch with 1 1/2 tablespoons cold water until smooth.
Open the slow cooker. If the chicken is done (165°F internal temperature), remove it with a slotted spoon and set aside; if not, leave it in.
Stir the cornstarch slurry into the liquid in the slow cooker, cover, and cook on HIGH for 20 minutes to thicken the sauce.
Return the chicken to the crock pot if you removed it, warm through briefly, then serve with rice if desired.
