Heat the oil in a Dutch oven or other large heavy pot over medium-high heat. Season the chicken with 1/2 teaspoon salt and the black pepper. Add the chicken in a single layer and cook for 4 to 5 minutes, stirring once or twice, until lightly browned. Transfer it to a plate; it does not need to be cooked through at this point.
Reduce the heat to medium. Add the onion and cook for 4 to 5 minutes, stirring often, until softened. Stir in the garlic, ginger, bell pepper, smoked paprika, and cayenne. Cook for 1 minute, just until fragrant.
Add the crushed tomatoes, 2 cups of the broth, sweet potato, remaining 1/2 teaspoon salt, and browned chicken with any juices. Bring to a gentle simmer. Cover partially, lower the heat to maintain a steady simmer, and cook for 18 to 22 minutes, stirring occasionally, until the sweet potato is nearly tender and the chicken is cooked through.
In a measuring cup or bowl, whisk the peanut butter with the remaining 1 cup broth until completely smooth. Stir this mixture into the stew. Simmer uncovered for 8 to 10 minutes, stirring frequently along the bottom of the pot, until the stew is creamy and the sweet potato is tender.
Stir in the spinach and cook for 1 to 2 minutes, until wilted. Remove the pot from the heat and stir in the lime juice. Taste and adjust the salt, cayenne, or broth as needed. Serve warm over rice, with peanuts and cilantro if desired.