Prepare sauce: In a blender or food processor, place first 6 ingredients (water through vinegar) along with 2 teaspoons of the oil; process until just a few chunks remain; set aside.
Heat remaining 1 tablespoon of oil in a large skillet over medium-high heat. Sprinkle salt and pepper on both sides of chicken breast halves then add to the skillet and cook for 3 to 5 minutes per side or until golden on the outsides and juices run clear (reducing heat as necessary to prevent burning). Transfer to a plate and tent with foil to keep warm.
To the same skillet, add prepared tomato sauce; bring to a boil then reduce heat and simmer till heated through.
Arrange chicken on dinner plates; top with sauce and garnish with cheese.