Ingredients

Tagliatelle

1 ½ cups unbleached all-purpose flour, plus more for dusting
1 large egg, plus 3 large egg yolks, room temperature
2 teaspoons extra-virgin olive oil
2 tablespoons warm water
Semolina flour, for dusting (optional)

Porcini Butter Sauce

¼ cup extra-virgin olive oil, plus more for drizzling
4 tablespoons unsalted butter
4 cloves garlic, smashed and peeled
1 pound porcini mushrooms, halved lengthwise and cut crosswise into ¼-inch-thick slices (smaller mushrooms can be left halved)
Kosher salt
¼ teaspoon dried Calabrian-chile or red-pepper flakes, plus more for serving
½ cup coarsely chopped fresh parsley leaves
1 ounce Pecorino Romano, finely grated (⅔ cup)

Directions

Tagliatelle: Mound flour on a clean work surface or in a large bowl. Make a well in center. Add egg, yolks, oil, and warm water to well. Using a fork and stirring outward from center, gradually incorporate flour into egg mixture until a ragged dough forms.
Transfer to a lightly floured work surface and knead until dough is smooth and elastic, 6 to 8 minutes. (Or combine ingredients in a mixer bowl and knead with the dough-hook attachment 8 to 10 minutes.) Tightly cover with plastic wrap and let stand at room temperature until soft and pliable, at least 30 minutes and up to 2 hours.
Cut dough into 4 portions. Keeping remaining portions covered with plastic, shape one into a rectangle. With a pasta roller on its largest setting (1 on a KitchenAid stand-mixer attachment), run dough through roller. Fold dough in half and run through largest setting again. Repeat once or twice more. Turn machine to next setting and pass through roller once.
Continue passing dough through, lowering setting each time, until dough has run through the smallest setting (8 on KitchenAid attachment). Transfer to a tray generously dusted with semolina (or all-purpose) flour; dust top with more. Repeat process with remaining portions of dough.
Attach a tagliatelle/fettuccine cutter to a pasta machine. Roll each sheet through cutter. Hang in a single layer on a pasta rack or transfer to a tray dusted with semolina and let dry, 30 minutes. (Pasta can be made up to 1 day ahead: Once dry, loosely twirl handfuls of pasta into 6 "nests," generously dust with more semolina, and refrigerate in a single layer in an airtight container.)
Porcini Butter Sauce: Heat a large straight-sided skillet over medium. Add oil, butter, and garlic. When butter melts and garlic sizzles, add mushrooms; season with salt. Cook, stirring occasionally, until mushrooms are tender and golden brown, 12 to 15 minutes. Add pepper flakes and cook 30 seconds more.
Meanwhile, cook pasta in a large pot of generously salted boiling water until al dente,
2 to 3 minutes. Reserve 1 cup pasta water; drain. Add 2/3 cup pasta water to skillet with mushrooms. Bring to a simmer. Add tagliatelle, tossing until sauce thickens slightly and evenly coats pasta, about 1 minute. Remove from heat.
Add parsley and half of cheese, tossing to combine. Add more pasta water as needed, a little at a time, until sauce evenly clings to pasta again. Serve topped with remaining cheese, pepper flakes, and a drizzle of oil.