1. Prepare the dough. Knead the flour on a work surface and form a mound; in the center, put the eggs, oil, and salt; mix the ingredients by kneading until the dough becomes soft and elastic. Wrap in plastic wrap and let rest for at least 30 minutes.
2. After this waiting time, roll out the dough with a rolling pin (or a food mill) into a thin layer; roll it up, and with a very sharp knife, cut into fettuccine strips.
3. Prepare the black olive sauce. Put the butter in a bowl, break it into pieces, and let it reach room temperature. Meanwhile, pit the olives, rinse them separately in cold water, drain, and dry on a clean kitchen towel. Chop finely by hand or with a food processor.
4. Rinse the capers under running water; drain, dry, and chop finely. Work the butter with a wooden spoon until it is soft and fluffy; add the chopped olives, chopped capers, grated lemon zest, extra virgin olive oil, lemon juice, and a pinch of salt and pepper. Mix vigorously until the ingredients are well combined and creamy.
5. In a large quantity of salted water, cook the pasta. Add 3 c. of the cooking water to the olive cream mixture and mix well. Drain the fettuccine when they are al dente, top with the black olive sauce, and serve.