Prep the chicken: Pat the chicken dry with paper towels.
For breasts, slice horizontally to make cutlets or pound to even thickness (about 1/2 inch). Season both sides with salt, pepper, paprika, and garlic powder.
Heat the pan: Set a large skillet (cast-iron or stainless) over medium-high heat. Add olive oil and 1 tablespoon butter.
When the butter foams, you’re ready to sear.
Sear the chicken: Place chicken in the pan without crowding. For thighs, skin-side down first. Sear 4–5 minutes per side for breasts, 5–7 minutes per side for thighs, until golden.
Remove to a plate. Thighs may need to finish in the oven later.
Lower heat and build the sauce: Reduce heat to medium-low. Add another tablespoon butter.
Stir in garlic and cook 30–60 seconds until fragrant, not browned.
Deglaze and season: Pour in chicken broth, scraping up brown bits. Stir in lemon zest, 2 tablespoons lemon juice, Dijon (if using), and capers. Simmer 2–3 minutes to slightly reduce.
Finish the sauce: Swirl in the last tablespoon of butter to make the sauce glossy.
Taste and adjust with more lemon juice, salt, or pepper.
Return chicken to the pan: Nestle chicken back into the sauce. Spoon sauce over the top. For breasts, simmer gently 2–3 minutes until cooked through.
For thighs, transfer the skillet to a 375°F (190°C) oven for 8–12 minutes, or until the thickest part reaches 165°F (74°C).
Rest and garnish: Let the chicken rest 3–5 minutes. Spoon sauce over again and finish with parsley and lemon slices.
Serve: Pair with mashed potatoes, rice, or a simple arugula salad. Don’t forget crusty bread to catch the sauce.