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Preheat the broiler, setting a rack about 4 or 5 inches from the flame (optional - for crisping the chicken if you'd like at the end).
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Rub all sides of the chicken with plenty of salt and fresh pepper. Finely dice the piperade, or puree it in a blender or food processor.
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Heat a large cast iron or steel pan over high heat until hot, at least 5-10 minutes.
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Add oil, and when it begins to shimmer (almost immediately), carefully add the chicken, skin side down. Do not overcrowd the chicken; cook in batches of necessary. Cook until skin browns, about 3 or 4 minutes. Carefully turn the chicken to the other side with tongs and cook for an additional 3 minutes, until lightly browned. Transfer chicken to a plate and discard the oil.
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Return the pan to medium high heat and add the brown sugar, whisking in the pan until it melts. Whisk in the sherry vinegar and heat for a minute or two, until it has thickened and reduced a bit.
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Stir in the piperade and the espelette pepper or red pepper flakes. Return the chicken to the pan, cover, decrease the heat to medium-low, and cook for 8-10 minutes, until the chicken is cooked through.
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For extra crispiness, transfer the chicken, skin side up, to a broiling pan and broiler for 2-3 minutes, or until the skin is crisp.
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Spoon some sauce on several plates and top with one or two pieces of chicken then more sauce. Garnish with parsley.