Special equipment: a pasta machine
For the homemade tagliatelle: Mound the all-purpose flour and salt in the center of a large wooden board; use your hands to make a well in the center of the mound. Whisk together the eggs and olive oil in a bowl, then pour into the well. Use a fork to slowly mix the eggs into the flour until they are completely incorporated.
Knead the pasta dough for about 8 minutes, adding flour in small increments if the board gets sticky. If the dough feels too dry, add 1 teaspoon of water occasionally as you go; the dough should feel elastic and smooth. Shape the dough into a ball and wrap in plastic. Let rest for 30 to 60 minutes at room temperature.
Set a pasta machine to its widest setting. Divide the dough into 4 pieces. Roll each piece through the machine repeatedly, working from the widest setting to the second thinnest. Cut the pasta sheets into 10-inch pieces, then run through the machine using the tagliatelle attachment.
Gather the tagliatelle in your hands, shake loose so the strands don't stick together, and then toss with some semolina flour. Divide into separate serving portions on a baking sheet.
For the tagliatelle with mushrooms: Bring a large stockpot of salted water to a rolling boil.
Meanwhile, heat the olive oil in a large skillet over medium-high heat. Add the mushrooms and cook until softened, about 5 minutes. Season with salt and pepper. Add the butter and mint; cook until the butter is melted and the mint begins to crisp, about 3 minutes. Remove from the heat.
Cook the pasta until al dente; strain and add to the sauce along with the 1/4 cup pecorino. Toss to coat the pasta in the sauce. Finish with a sprinkling of pecorino and freshly ground black pepper.