In a large bowl, whisk together the vinegar and 3 tablespoons of the oil. Season with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper. Add the celery, onion, olives, and sprouts and toss well.
Rinse the chicken under cold water and pat dry with paper towels. Place between 2 pieces of plastic wrap and pound with a mallet, rolling pin, or bottom of a skillet to an even 1/2-inch thickness. Season the chicken with the remaining salt and pepper.
Heat the remaining oil in a skillet over medium heat. Working in batches, add the chicken and saute until cooked through, about 3 minutes per side. Transfer the chicken to plates and top with the salad.
Tip: Pounding the chicken ensures that it will cook quickly and evenly. For a time-saving alternative, look for thinly sliced cutlets, available at many grocery stores. Because the pieces are smaller than standard breasts, allow 2 per person.