Browned butter is the quick cook's best-kept secret: It takes less than 2 minutes to caramelize the milk solids in butter for a fragrant, nutty note in any dish.
Place chicken breasts on a cutting board; pound to a 1/2-inch-thickness using a meat mallet or small, heavy skillet (all four breasts should fit in one large skillet). Sprinkle chicken with 1/2 teaspoon salt and 1/2 teaspoon pepper.
Heat 1 tablespoon oil in a large skillet over medium-high. Add chicken to pan; cook 4 to 5 minutes on each side or until done. Remove from pan; keep warm. Do not wipe pan clean.
Reduce heat to medium. Add remaining 1 tablespoon oil, remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, butter, and garlic to drippings in pan; cook 2 minutes or until butter just begins to brown, stirring frequently. Stir in tomatoes; cook 2 minutes or until tomatoes are wilted. Spoon tomato mixture over chicken; sprinkle with parsley.