Heat 2 tablespoons olive oil in a stock pot over medium heat. Add shallot, mushrooms, sage and garlic. Cook for 8 minutes, until mushrooms give off their liquid.
Add 4 cups broth and potatoes, and bring to a boil over medium-high heat. Reduce heat to medium, and simmer for 10 minutes.
While soup is simmering, heat remaining 1 tablespoon olive oil in a small skillet over medium heat. Add sage leaves, and cook until just crisp, 3-4 minutes. Remove to a paper towel, and reserve oil.
When potatoes are tender, coarsely puree soup with an immersion blender (or—carefully—in batches in a food processor). Stir in beans and remaining 2 cups broth, season with salt and pepper, and bring back to a simmer until hot.
To serve, drizzle with sage oil, and top with fried sage leaves.