Season skin sides chicken with salt, black pepper, and cayenne pepper; flip and season skinless sides with salt, black pepper, cayenne pepper, and herbes de Provence. Flip again so skin sides facing up; rest 10 minutes.

Heat olive oil in a large skillet over high heat. Add chicken skin-side down; reduce heat to medium-high and cook chicken until skin well-browned, about 5 minutes. Flip and cook until just browned, about 2 minutes; transfer to a plate.

Reduce heat to medium. Add shallots; cook and stir until just softening and golden brown, about 2 minutes. Stir in olives; cook until fragrant, about 30 seconds.

Add chicken broth; bring to a boil while scraping browned bits off bottom of the skillet with a wooden spoon. Add 2 tablespoons reserved olive brine, lemon juice, lemon zest, and cumin; increase heat to high and cook until sauce is reduced by half, 2 to 3 minutes.

Reduce heat to low; return chicken, skin-side up, to the skillet. Cover the skillet; cook until chicken 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). Transfer chicken to a plate; cover with aluminum foil.

Stir parsley and butter into sauce, stirring constantly, until butter melted and sauce shiny. Spoon sauce over chicken.