Season the chicken with 1 teaspoon salt and a couple turns of black pepper. Heat a Dutch oven over medium-high heat. When hot, pour in 2 tablespoons vegetable oil. Add the chicken and cook, turning often, until brown on all sides, about 6 minutes; transfer to a bowl and set aside.
Add the remaining 1 tablespoon vegetable oil to the pot and reduce the heat to medium. Add the onion, garlic, ginger and 1 teaspoon salt. Cook until the onion is softened, about 5 minutes. Add the sweet potato, apple, cayenne, chicken broth and 3 cups water; bring to a simmer. Stir in the kale, tomato and almond butter and simmer until the vegetables are tender, 10 to 15 minutes.
Return the chicken to the pot and simmer until just cooked through, about 5 minutes. Serve topped with the cilantro.