Heat the oil in a large skillet over medium high heat. Season the chicken breasts with salt and pepper and cook for about 4 or 5 minutes on each side, until cooked through. Lower the heat if the outside of the chicken is getting too brown. Remove the chicken to a plate.
Return the skillet to medium heat, add the butter, and when the butter is melted add the garlic. Stier the garlic for 1 minute until it’s golden brown. Add the cilantro, lime juice and zest and stir, scraping the bottom, so that any browned bits are released and become part of the sauce. Return the chicken to the skillet, along with any juices that might have accumulated, and turn the chicken so that it is coated with the sauce and cilantro. Transfer the chicken to a serving plate, and pour over any sauce left in the pan, scraping to make sure you get all of the herbs and garlic.