Place the sweet potatoes in a large pot. Cover with water and bring to a boil over high heat. Cook until just barely tender, 4-5 minutes. Drain and set aside.

Heat 1 tablespoon of the oil in a large skillet over medium-high heat. (Use a cast-iron skillet if possible.) Add the sausage and cook, crumbling up with a utensil, until browned, 5-6 minutes. Remove the sausage from the skillet. Add the remaining 2 tablespoons of oil. Add the peppers and onion. Cook, stirring often, for 5 minutes. Add the garlic, cumin, and chili powder and stir for 1 minute. Add the kale, sausage, and sweet potatoes. Cook, stirring occasionally, for about 5 minutes. Season to taste with salt and pepper. Serve with salsa on the side.