1 ounce dried porcini mushrooms
3 pounds 8 ounces mixed fresh mushrooms, such as cremini, shiitake, hen of the woods, and oyster
4 tablespoons extra-virgin olive oil
2 medium onions, minced (see note above)
1 large carrot, minced (see note above)
15 medium cloves garlic, minced (see note above)
1 cup dry white wine
1 (15-ounce) can whole peeled tomatoes, crushed with juices
4 sprigs thyme
Kosher salt and freshly ground black pepper
In a medium heatproof mixing bowl, combine dried porcini with 2 cups boiling water and let stand 15 minutes. Meanwhile, wash fresh mushrooms and discard any woody stems, such as on cremini and shiitake mushrooms (or save stems for making mushroom or vegetable stock).
Thinly slice mushrooms (in the case of hen of the woods mushrooms, tear by hand into thin strips).
Lift rehydrated dried mushrooms from soaking liquid and squeeze to remove excess liquid; fine-strain soaking liquid and reserve.
In a large pot, heat oil over high heat until shimmering. Add onion, carrot, and garlic, and cook, stirring, until softened and beginning to brown on bottom of pot, about 6 minutes.
Add fresh and soaked dried mushrooms and cook, stirring and scraping bottom of pot, until they release their water and begin to brown on bottom of pot, about 10 minutes. Continue cooking, scraping bottom of pot and making sure to reach into corners, until it becomes difficult to scrape up browned bits and they threaten to burn, about 15 minutes longer.
Add wine, scraping up browned bits on bottom of pot. Add tomatoes, dried-mushroom soaking liquid, and thyme.
Lower heat to medium-low and cook, stirring often and scraping bottom of pot to prevent scorching, until mushroom sauce is thick, 1-2 hours. Discard thyme stems. Season with salt and pepper.
Mushroom ragù can be served on pasta or with polenta.
Keep refrigerated for up to 5 days or freeze for up to 6 months.