Trim chicken of excess fat; season with salt and pepper.
Melt butter with oil in a skillet over medium-high heat. Add chicken; cook, turning often, until lightly browned, about 10 minutes total.
Remove chicken to a plate; pour off all but 1 Tbsp. fat. Sauté onion until softened, about 5 minutes. Add broth and cook, stirring to pick up browned bits of chicken on bottom of skillet. Stir in peanut butter and 1/4 cup water. Cook, stirring, until sauce is smooth. (If sauce is too thick, add water, 1 Tbsp. at a time.) Stir in red pepper.
Return chicken to skillet; turn to coat. Cover, reduce heat to medium-low; cook until chicken is cooked through, 25 minutes. Remove chicken to platter. Skim fat from sauce, stir in lime juice and 1 Tbsp. cilantro; spoon over chicken. Sprinkle with remaining cilantro.