Heat olive oil in a large (5.5-quart) Dutch oven or pot over medium-high heat. Add the mushrooms and sauté, stirring frequently, until tender, about 5 minutes.
Add Parmigiano Reggiano and let it sit on the bottom of the pot, allowing the cheese to brown and form a fond. Stir every 15 to 20 seconds to prevent burning.
Stir in onion, leek greens, carrots, celery, parsley, bay leaves, and peppercorns. Cook until the vegetables are slightly soft and tender, 6 to 7 minutes.
While scraping the browned bits off the bottom of the pot, slowly pour ½ cup of water to deglaze. Add the remaining water, reduce the heat to medium-low and simmer for 50 minutes to 1 hour, stirring occasionally.
Strain into a container, cool, and refrigerate for up to 1 week or store in the freezer for up to 3 months.
