Rinse and pat the chicken dry. If breasts are thick, slice them horizontally to an even thickness (about 1/2") so they cook evenly.
Make the marinade: in a food processor or blender combine cilantro (stems and leaves), lime juice, lime zest, garlic, olive oil, cumin, salt, cayenne and black pepper. Pulse until a slightly chunky, spoonable sauce forms — scrape down the sides once or twice and pulse again.
Transfer chicken to a large zip-top bag or shallow dish. Use a fork to prick the surface of the chicken a few times (this helps the marinade penetrate). Pour in the cilantro-lime mixture, remove excess air if using a bag, and press the marinade around the pieces so everything is coated.
Marinate in the refrigerator for at least 30 minutes and up to 24 hours. For quickest flavor, let sit 30–60 minutes; for deepest flavor, refrigerate overnight.
Pan-fry method: heat a large nonstick or cast-iron skillet over medium-high heat and add 1 teaspoon oil to coat. Remove chicken from the marinade (discard any leftover marinade). Sear chicken 4–5 minutes without moving to develop a golden crust, flip, then reduce heat to medium-low and cook another 4–6 minutes until the internal temperature reaches 160°F (it will rise while resting).
Grill method: preheat grill to medium-high (about 450°F). Oil the grates. Place chicken on the grill and cook 4–6 minutes per side depending on thickness, closing the lid between flips, until the internal temperature is 160°F.
Oven method: preheat oven to 425°F. Lightly oil a baking dish and lay chicken in a single layer. Bake 18–28 minutes (thinner pieces near 18 min; thicker up to 28 min) until the thickest part reads 160°F.
Once cooked, transfer chicken to a cutting board, tent loosely with foil, and let rest 8 minutes so the temperature evens out and juices redistribute (final temperature should reach 165°F). Slice and serve with extra cilantro leaves and lime wedges.