Pat the chicken dry and season it with the salt and black pepper. Heat the oil in a large Dutch oven or heavy soup pot over medium-high heat. Brown the chicken for 5 to 7 minutes, stirring occasionally, until lightly golden on a few sides. Transfer the chicken to a plate; it does not need to be cooked through at this stage.

Reduce the heat to medium. Add the onion to the pot and cook for 4 to 5 minutes, stirring often, until softened. Add the garlic and ginger and cook for 30 seconds, just until fragrant.

Stir in the crushed tomatoes and tomato paste. Cook for 2 minutes, scraping up any browned bits from the bottom of the pot. Return the chicken and any juices on the plate to the pot.

Pour in 3 cups of the chicken broth (720 mL). Add the sweet potatoes and cayenne, if using. Bring the stew to a gentle simmer, then lower the heat, cover partially, and cook for 18 to 20 minutes, stirring once or twice, until the sweet potatoes are nearly tender.

In a medium bowl or large measuring cup, whisk the peanut butter with the remaining 1 cup chicken broth (240 mL) until smooth. Stir the peanut mixture into the stew. Simmer uncovered for 8 to 10 minutes, stirring frequently to prevent sticking, until the sauce is creamy, the sweet potatoes are tender, and the chicken is cooked through.

Stir in the kale and cook for 2 to 3 minutes, until wilted. Taste and adjust the salt, pepper, or cayenne as needed. Stir in the lime juice, if using. Serve hot with cooked rice, couscous, or warm crusty bread.