Place chicken breasts between 2 pieces of plastic wrap. Pound with a rolling pin, meat mallet or heavy skillet until flattened to an even thickness, about 1/2 inch. Sprinkle with 1/4 teaspoon salt and pepper. Place flour in a shallow dish. Dredge the chicken in the flour, shaking off excess. (Discard any leftover flour.)

Heat oil in a large skillet over medium heat. Add the chicken and cook until browned and no longer pink in the center, 3 to 5 minutes per side. Transfer to a plate, cover and keep warm. (If necessary, cook the chicken in two batches with an additional 1 tablespoon oil.)

Off the heat, add wine and shallot to the pan. Return to medium heat and cook, scraping up any browned bits, until slightly reduced, about 3 minutes. Add apricots and cook until the fruit begins to break down, 2 to 3 minutes. Stir in preserves, tarragon and the remaining 1/2 teaspoon salt. Return the chicken to the pan and cook until heated through, 1 to 2 minutes. Serve the chicken with the sauce.