Heat a large skillet to medium heat. Add coconut oil, chicken pieces and cook for about 7 minutes (it's okay if they are still a little pink inside, but you want them mostly cooked through). Remove chicken to a plate.
Add 1 more teaspoon coconut oil, then add ginger and garlic (and peppers if using). Sauté for a minute. Add curry paste, coconut milk and use a whisk to combine.
Add carrots and chopped kale, stirring to combine. Next, add chicken broth, a pinch of salt and the juice from one lime. Simmer for about 5 minutes.
Next, add chicken back to pan and simmer for a few more minutes or until chicken is cooked through. Taste and add an additional pinch of salt if needed. Serve over basmati rice or rice noodles.