For the fresh pasta: Combine the flour, eggs and yolk in a food processor and pulse until a dough comes together. Transfer to a counter dusted lightly with flour and knead gently until the dough comes together and is smooth, about 1 minute. Wrap in plastic and refrigerate for at least 30 minutes and up to 24 hours.
Divide the dough into 4 pieces. With your hands, flatten and shape one piece of dough into a 1/2-inch-thick rectangle. Dust it lightly with flour and pass it through the widest setting on a pasta machine. (If the dough comes out oddly shaped, re-form into a rectangle.) Fold it into thirds, like a letter, and if necessary, flatten it to a 1/2-inch thickness. Pass it through the widest setting again with the seam of the letter perpendicular to the rollers. Repeat this folding and rolling step 10 to 12 times, dusting the dough with flour if it becomes sticky.
Without folding the dough, pass it through the next setting on the pasta machine. Continue to pass the dough through the machine, reducing the space between the rollers after each pass and lightly dusting the pasta with flour on both sides each time, until the pasta is about 1/16-inch thick and 3 inches wide.
Lay the rolled-out dough on a floured counter. Roll out the remaining dough in the same manner. Cut each piece of dough into 11-inch lengths. Cut the dough into fettuccine by hand, or by passing through the pasta machine on the fettucine setting.
For the black truffle sauce: Boil a large pot of salted water. Add the pasta and boil until al dente, about 3 minutes. Strain, reserving 2 cups of pasta water.
Add the butter to a large pan over medium heat and let melt. Add half the Parmesan and 1/2 cup pasta water and stir to combine. Add the pasta and toss to coat. Add the remaining Parmesan and stir to combine. If the sauce is too thick, add more pasta water. Divide pasta among 4 bowls. Top with shaved truffle and more Parmesan. Enjoy!