Sauté the mushrooms in the oil. This is a very small amount of oil for one pound of mushrooms. But if you keep the heat low, the mushrooms will release their own moisture and before you know it you'll have a bunch of water in the pan that won't allow them to burn. Just get the pan hot over high heat and then immediately turn to low heat. Stir frequently.
When the liquid comes out of the mushrooms, add the balsamic vinegar. Then, the flour, which will soak up all the liquid pretty quickly.
Put the chicken broth into your soup pot. Transfer the mushrooms to the same pot and combine with the broth.
Bring to a boil. Reduce heat and simmer for approximately 10 minutes.
Allow to cool in the pot. If you will be eating this, season to taste with salt and pepper. If you will be using this soup for other recipes, do not add salt at this time.
Please note that the nutrition data given here is a ballpark figure. Exact data is not possible.