Heat 2 tablespoons oil in a 10-inch skillet over medium-high heat. Add the chicken and cook for 10 minutes or until it's well browned on both sides. Remove the chicken from the skillet.
Heat the remaining oil in the skillet over medium heat. Add the shallot and cook and stir for 2 minutes. Stir the soup, water, tomatoes, vinegar and chopped basil in the skillet.
Return the chicken to the skillet and heat to a boil. Reduce the heat to low. Cook for 5 minutes or until the chicken is cooked through. Serve the chicken and sauce over the noodles. Sprinkle with the cheese and sliced basil, if desired.