Step 1

Bring 4 quarts water to boil in Dutch oven. Add pasta and 1 tablespoon salt and cook, stirring occasionally, until al dente. Reserve 1 cup cooking water, then drain pasta and return it to pot.

Step 2

Meanwhile, pulse mushrooms in food processor until finely chopped, about 10 pulses. Heat oil in 12-inch skillet over medium-high heat until just smoking. Add mushrooms, onion, carrot, and salt and cook until mushrooms appear dry and begin to stick to bottom of skillet, about 14 minutes.

Step 3

Stir in tomato paste and garlic and cook until fond forms on bottom of skillet, about 1 minute. Stir in wine, scraping up any browned bits, and cook until evaporated, about 2 minutes. Stir in reserved cooking water and bring to boil. Add sauce and Pecorino to pasta in pot and stir to combine. Serve with extra Pecorino.