Heat 2 tablespoons of the olive oil in a large skillet over medium heat. Add the onions and saute for 3 to 5 minutes until translucent, then add the garlic, thyme, tarragon, salt, and papper, and cook another minute. Next, add the mushrooms and cook for 7 to 10 minutes until they are very soft, lowering the heat if necessary to prevent them from burning.
While the mushrooms are cooking, place the walnuts in a food processor or blender and process until very fine.
Add the cooked mushroom mixture to the walnuts in the food processor, along with the balsamic vinegar, beans, and remaining tablespoon of olive oil. Process until smooth, adding the vegetable broth 1 tablespoon at a time as needed. Continue to puree the ingredients until the pate resembles a smooth, thick and spreadable paste. Scrap mixture into an airtight container and chill for at least an hour before serving, to allow the flavors to meld.