In a large bowl, whisk the vinegar, 3 tablespoons oil, ¼ teaspoon salt, and ¼ teaspoon pepper. Add celery, onion, olives, and sprouts. Toss well.
Rinse chicken under cold water; pat dry with paper towels.
Between 2 pieces of plastic wrap, pound chicken to ½-inch thickness with a mallet or the bottom of a skillet. Season with remaining salt and pepper.
Heat the remaining oil in a skillet over medium heat. Working in batches, sauté the chicken until cooked through, about 3 minutes per side.
Transfer to plates and top with the salad.