Place chicken breast in casserole dish, cover with balsamic vinegar and place in oven to bake.
Heat olive oil in a large skillet over medium-low heat. Carmelize the onion, (saute the onion and garlic until the onion is light brown, 10 minutes). In the meantime heat the water and vegetable stock in a small pot or fry pan to a light boil. Turn off the heat. Add salt, pepper and mushrooms to the onions, and cook until the mushrooms are soft.
Pour the milks into the skillet, and heat for 2 minutes. Stir in the barley. Stir in the vegetable stock/water mixture one cup at a time. Each new cup being added when the last has been absorbed. Stir as needed.
When the chicken is finished cooking, cut into small slices and add to rice mixture with the last cup of water/stock. When the rice has finished cooking, (45 minutes or so) add the butter and cheeses. Serve hot.