Combine first 3 ingredients in a small bowl. Rub miso mixture over both sides of chicken breast halves. Wrap each breast half securely in plastic wrap. Arrange the packets in steamer rack; place rack in a Dutch oven. Steam packets, covered, 20 minutes or until done. Remove packets from steamer; let stand 5 minutes. Remove chicken from packets, reserving liquid from packets. Dice chicken; set aside. Discard water in pan; wipe pan dry with a paper towel.
Place a large nonstick skillet coated with cooking spray over medium-high heat until hot. Add egg whites, and cook 2 minutes or until done. Remove egg whites from skillet; coarsely chop.
Add reserved cooking liquid, chicken, onion, carrot, and fish sauce to Dutch oven, and bring to a boil. Reduce heat to medium; cook 5 minutes or until liquid almost evaporates. Add rice, mushrooms, parsley, green onions, and soy sauce; cook 3 minutes. Stir in egg whites and spinach.