Pour the flour onto a work surface and shape into a mound. Form a deep, wide well in the center and add the eggs; sprinkle with the salt.
Beat the eggs lightly with a fork, then gradually work in the flour, moving from the inside of the well in a circular motion.
Set the dough aside, clean and flour the work surface, then knead the dough until smooth and elastic, about 10 minutes.
Shape the dough into a ball, dust with more flour and wrap with plastic wrap. Let rest for at least 30 minutes.
Divide the dough into 6 equal pieces and cover. On a floured work surface, flatten 1 dough piece. Using a pasta machine, feed the dough through the widest setting, then feed through the same setting 3 more times. Change to the second-widest setting and feed the pasta sheet through twice. Repeat this pattern until you reach the thinnest setting, lightly dusting with flour as necessary, then feed the pasta sheet through once; lay the pasta sheet on a lightly floured kitchen towel. Repeat with the remaining dough pieces.
Let the pasta sheets dry, about 10 minutes. Set the pasta machine to the flat noodle or "tagliatelle" setting and feed the sheets through the machine. Transfer the cut pasta to kitchen towels.
Bring a large pot of salted water to a boil. Add the pasta and cook until al dente, 3 to 5 minutes.