Sauté vegetables: Heat the olive oil in a large pot or Dutch oven over medium heat. Add the chopped onion, diced carrots, and diced celery. Cook for about 5-7 minutes, stirring occasionally, until the vegetables are softened.
Add mushrooms: Add the sliced cremini and shiitake mushrooms to the pot. Cook for about 8-10 minutes, stirring occasionally, until the mushrooms have released their liquid and browned slightly.
Add garlic and spices: Stir in the minced garlic, dried thyme, dried rosemary, and red pepper flakes (if using). Cook for 1 minute more, until fragrant.
Deglaze with wine: Pour in the white wine and scrape up any browned bits from the bottom of the pot. Bring the wine to a simmer and cook for a few minutes, until the wine has reduced slightly.
Add broth and beans: Add the vegetable broth, cannellini beans, and Great Northern beans to the pot. Bring the mixture to a simmer, then reduce the heat to low and simmer, covered, for 20-25 minutes, stirring occasionally, to allow the flavors to meld.
Finish and serve: Stir in the chopped parsley. Season the stew with salt and pepper to taste. Serve hot, garnished with a drizzle of olive oil and with crusty bread for dipping, if desired.