1. Heat oil over high heat for 3 to 4 minutes in a deep pot or large heavy skillet. Add chicken pieces, half at a time, and fry on both sides until nicely browned. Once all chicken pieces are cooked, set aside in a covered bowl to keep warm.
2. Add onions and garlic in same pot; cook until onions are soft and begin to brown, stirring occasionally. Add pureed tomatoes and 1 cup water; reduce heat to medium, bring to a simmer, and let simmer for a few minutes.
3. Return chicken to pot. Stir in peanut butter. Mixture will look clumpy for a minute, but will mix in nicely once it heats up. Add cayenne pepper, black pepper, and salt. Simmer over low heat 10 to 15 minutes.
4. Stir in diced green pepper and tomatoes; simmer for 3 to 4 more minutes.