If you don't have time to dry brine: Pat chicken dry with paper towels. Sprinkle chicken with 1 ½ teaspoons salt and ¼ teaspoon pepper. If you have time to dry brine: Set a wire rack over a rimmed baking sheet and arrange chicken breasts on it. Sprinkle all over with 1 ½ teaspoons salt and ¼ teaspoon pepper, then refrigerate, uncovered, for at least 30 minutes or up to 12 hours.

Brush chicken breasts all over with 2 tablespoons oil. In a 12-inch nonstick, carbon-steel, or cast iron skillet place prepared breasts, arranging so narrow ends are opposite wider ends. Place skillet over medium-high heat and cook for 3 minutes. Flip breasts and cook on second side until lightly browned on bottom, 3 minutes.

Flip breasts; reduce heat to medium; and continue to cook, flipping every 2 minutes, until exterior is well browned and thickest part of chicken registers 155℉ (68℃), 8 to 12 minutes longer. Transfer chicken to platter; let rest while preparing bulgur.

To now-empty skillet, add remaining 1 tablespoon oil, tomatoes, and scallion whites and cook until tomato skins blister and begin to release their juices and scallion whites are softened, 1 to 2 minutes. Stir in 1 ½ cups water and bring to a boil, scraping up any browned bits. Stir in bulgur and remaining ¾ teaspoon salt. Bring to a boil, then cover, remove from heat and let sit for 5 minutes.

Fluff bulgur with fork and stir in feta, olives, and remaining 2 tablespoons oil. Season with salt and pepper to taste. Slice chicken and serve with bulgur pilaf. Sprinkle with scallions greens and serve with lemon wedges, if using.