Step 1

In a large (about 8-quart) pot, heat the oil over medium-high until just shimmering. Add the mushrooms and 1 teaspoon salt; cook, stirring occasionally, until well browned, about 5 minutes. Add the tomato paste and cook, stirring, until it begins to brown and stick to the pot, about 1 minute. Add 14 cups water, scraping up any browned bits. Add the onion, carrots, celery, tomatoes, potatoes, parsley, bay, thyme and peppercorns. Bring to a boil over high, then reduce to medium-low and simmer, uncovered and stirring occasionally, for 1 hour.

Step 2

Remove the pot from the heat. Cool for 15 minutes, then strain through a fine-mesh strainer set over a large bowl; press firmly on the solids to extract as much liquid as possible; discard the solids. You should have 2 quarts strained broth; if you have less, add water to reach that amount. Let the broth cool completely, then cover and refrigerate for up to 5 days or transfer to an airtight container and freeze for a few months.