Season the chicken. Pat the chicken dry and season with 1 teaspoon salt and 1/2 teaspoon black pepper.
This helps the meat stay flavorful as it cooks.
Bloom the spices. In a skillet over medium heat, warm the oil. Add the onion and cook 3–4 minutes until softened. Stir in garlic and ginger for 30 seconds, then add curry powder, cumin, coriander, turmeric, and cayenne.
Cook 1 minute, stirring, until the spices smell toasty.
Build the base. Add tomato paste to the skillet and cook 1 minute. Stir in the diced tomatoes and broth, scraping up any browned bits. Bring to a gentle simmer, then remove from heat.
Layer the slow cooker. Add sweet potatoes and chicken to the slow cooker.
Pour the tomato-spice mixture over the top. Stir to coat everything evenly.
Set and cook. Cover and cook on Low for 6–7 hours or High for 3–4 hours, until the chicken is tender and the sweet potatoes are soft but not falling apart.
Finish with coconut milk. Stir in the coconut milk and brown sugar. Let it warm through on Low for 10–15 minutes.
Taste and adjust salt, then add lime juice to brighten the flavors.
Serve and garnish. Spoon over rice or serve with naan. Top with chopped cilantro and a squeeze more lime if you like.