Preheat an oven to 180 degrees C. Place chicken breasts in a baking dish and coat with balsamic vinegar.
Bake the chicken breasts in the preheated oven until no longer pink in the centre and the juices run clear; about 45 minutes. Slice chicken into small pieces. Set aside.
In the meantime; bring the water and vegetable stock to a low boil in a saucepan then turn off the heat. Heat the olive oil in a large heavy-bottomed saucepan over medium-high heat. Add the onion and garlic; cook and stir until the onion begins to turn golden brown at the edges; about 2 minutes.
Season with salt and pepper then add mushrooms cooking until mushrooms are tender. Pour the milk and the evaporated milk into the same saucepan and heat for 2 minutes. Stir in the barley.
Reduce the heat to medium and stir in 1 cup of the boiling vegetable stock; continue stirring until the barley has absorbed the liquid and turned creamy. Repeat this process until there is only 1 cup of stock left.
Stir in the chicken with the last cup of stock. When finished the barley should be tender yet slightly firm; about 45 minutes.
Stir in the butter, Parmesan cheese and Romano cheese before serving.