Make a mound of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs and oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour starting with the inner rim of the well. As you expand the well, keep pushing the flour up to retain the well shape. Do not worry that this initial phase looks messy. The dough will come together when half of the flour is incorporated. Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up any leftover crusty bits. Discard these bits. Lightly flour the board and continue kneading for 3 more minutes. The dough should be elastic and a little sticky. Continue to knead for another 3 minutes, remembering to dust your board when necessary. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature. Note: Do not skip the kneading or resting portion of this recipe. They are essential for a light pasta. Roll out pasta to thinnest setting on pasta machine. Cut pasta into 1/4-inch thick noodles by hand or with machine and set aside under a moist towel.
Follow regular fettucine instructions.
To Prepare Dish: Bring 6 quarts water to boil and add 2 tablespoons salt. In a 12 to 14-inch saute pan, heat oil over the heat. Add shallots and saute until light golden brown, about 5 to 6 minutes. Add radicchio and cook 30 seconds. Add cream, reduce by half and remove from heat. Drop pasta into boiling water and cook until tender yet al dente. Drain pasta well and toss into pan with cream mixture. Return to heat, add lemon zest and juice and toss to coat. Add half the cheese and toss again. Serve immediately in warm bowl, making extra cheese available on the side.