For the pasta, sift flour and salt together onto a large work surface. Make a well in the center of the flour, add eggs, one at a time, working into flour with your hands. Continue kneading until dough is smooth and elastic, about 10 minutes.
Divide dough into 4 balls. Roll each out as thinly as possible—either on a lightly floured work surface using a rolling pin or with a pasta machine. Cut each sheet into 12 to 14 4'' squares. Set squares aside, layered between clean dish towels.
Bring a large pot of salted water to a boil over high heat. Meanwhile, pulse garlic and pine nuts with salt in a food processor until finely ground. Add basil and pulse while slowly drizzling in oil. Add pecorino sardo and parmigiano-reggiano and purée, frequently scraping sides with a spatula, until pesto is smooth and creamy. Set aside.
To the boiling water, add 1 tbsp. oil and half the pasta, and cook until pasta is tender, 3–4 minutes. Remove carefully with a slotted spoon, drain in a colander, and toss with half the remaining oil. Repeat with remaining pasta. Dilute pesto with ½ cup of pasta cooking water and purée until creamy, about 1 minute. Gently toss hot pasta with pesto and serve immediately.