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1
Season skin-side of chicken with salt, black pepper, and cayenne pepper. Turn breasts over and season skinless side with salt, black pepper, cayenne, and herbes de Provence. Turn again so that the skin-side is facing up; rest for 10 minutes.
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2
Heat olive oil in a large skillet over high heat. Place chicken skin-side down in hot oil. Reduce heat to medium-high and cook chicken until skin is well-browned, about 5 minutes. Turn chicken and cook until just browned, about 2 minutes; transfer to a plate.
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3
Reduce heat to medium. Cook and stir shallots in the hot skillet until just softening and golden brown, about 2 minutes. Stir olives into shallots and cook until fragrant, about 30 seconds.
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4
Pour the chicken broth into the skillet; bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Add lemon juice, 2 tablespoons of the reserved olive brine, lemon zest, and cumin; increase heat to high and cook until sauce is reduced by half, 2 to 3 minutes.
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5
Reduce heat to low and return chicken, skin-side up, to the skillet. Cover the skillet with a lid and cook until no longer pink in the center and the juices run clear, about 5 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove chicken to a plate and cover with aluminum foil.
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6
Stir parsley and butter into sauce, stirring continuously until butter is melted and sauce is shiny. Spoon sauce over chicken.
