Heat 1 Tbsp oil in 12 in dutch oven over med-high heat. Add zucchini and saute until slightly browned, about 3-5 min., remove and place in bowl.
Add 1 Tbsp oil and saute mushrooms until they are soft, about 3-5 min., remove and place on plate.
Coat chicken with flour. Add 2 Tbsp oil and cook until both sides are lightly browned and the juices run clear when pierced by a fork, about 10 min. Remove to plate.
Cook the onion in pan drippings until tender, about 5 min. Stir in the water, add the chicken, mushrooms, tomatoes, balsamic vinegar and salt. Reduce heat to low and simmer uncovered 5 min.
Add zucchini and heat through.