Set pressure cooker to SAUTE. Add butter, mushrooms, onions, and garlic to the pot and stir once butter is melted. Sprinkle in flour and stir everything to incorporate flour (it should coat the other ingredients in a paste-like coating).
Stir in marsala cooking wine and simmer until thickened. Add chicken, chicken broth, and balsamic vinegar. Cover and set to PRESSURE COOK or MANUAL for 10 minutes.
Do a quick release (turn steam valve to VENT and allow to depressurize until float valve drops, then remove lid). Set to SOUP setting. Remove chicken, shred with two forks, and return to pot.
Once soup is boiling, add noodles and cook about 7 minutes or until noodles are tender. Add salt and pepper to taste, garnish with grated parmesan cheese if desired, and serve.
Complete step 1 above in a skillet on the stove over medium heat. Stir in cooking wine and simmer until thickened. Add to slow cooker along with chicken, chicken broth, and balsamic vinegar. Cover and set to HIGH for 2 hours or LOW for 4 hours.
Remove chicken, shred with two forks, return to pot along with noodles. Cook 20-30 minutes on HIGH until noodles are tender. Season with salt and pepper to taste, and garnish with grated parmesan cheese if desired before serving.