In a large resealable plastic bag, add the lemon zest and juice, oregano, 1 1/4 teaspoons salt and 1 tablespoon of the oil. Massage the bag to combine the ingredients. Add the chicken breasts, seal the bag and massage the marinade evenly into the chicken. Lay the bag flat with the chicken breasts all flat in the bag and let the air out of the bag. Using a small skillet, flat mallet or even the heel of your hand, pound the chicken to flatten slightly. It should be 1/2 to 3/4 inch thick. Marinate in the refrigerator for at least 4 hours and up to 10 hours.
Remove the chicken from the refrigerator 15 minutes before cooking.
Heat a large skillet over medium-high heat. Add the remaining 1 tablespoon oil to the hot skillet and swirl to coat the bottom. Add the chicken, smooth side down, and place something heavy on top of the chicken (a brick wrapped in aluminum foil or small cast-iron skillet works well here). Cook until deep golden brown on the first side, about 5 minutes. Remove the weight and flip the chicken. Continue to cook on the second side until it is firm to the touch and an instant-read thermometer inserted in the thickest part reads 160 degrees F, about 5 minutes more. Remove to a plate to rest. Add 1/3 cup of water to the skillet to deglaze. Pour the juices over the rested chicken to serve.