Sauté mushrooms and onion in olive oil over medium-high heat in a dutch oven (or regular pot) until some of the water from the mushrooms evaporates (about 5 minutes).
Add celery and carrots and cook another 5 minutes.
Add garlic and red wine and cook until the smell of alcohol is gone (about 5 minutes or so).
Add the potatoes, tomatoes, broth, Italian seasoning, salt, pepper, parsley, and red pepper flakes and bring to a boil.
Cover and reduce heat to simmer for about an hour, or until potatoes are tender.
Mix the water and flour in a separate small bowl to make a paste.
Add peas, navy beans, and flour mixture to the pot. Increase heat to medium or medium-high to bring back up to a slow boil and cook uncovered for another 20 minutes.
Add Parmigiano-Reggiano cheese to serve.