In a medium bowl, mix 2 tablespoons of vinegar, mustard, and garlic.
Add chicken and turn to coat.
Transfer chicken and marinade to skillet.
Saute chicken until cooked through, about 3 minutes on each side.
Transfer chicken to a platter and keep warm.
Prepare linguine according to package directions.
Melt remaining teaspoon of butter in the same skillet.
Saute onions and peppers for a few minutes, until they begin to soften.
Add the cabbage and saute for one minute.
Add mushrooms and saute for an additional minute.
Add broth, thyme, and remaining 4 tablespoons of vinegar.
Cook, stirring occasionally, until mushrooms are deep brown, about 2 minutes.
To serve, place a small chicken breast (or a half of a breast) on the plate. Add an equal amount of whole wheat linguine. Top the linguine with veggies, and spoon some of the balsamic sauce (now called a reduction because you cooked some of the water out of it) over the chicken, noodles, and vegetables.
Notes
*If your chicken breasts are very thick, cut them in half lengthwise. They cook quicker and turn out much juicier if they are thinner.