Place flour on a marble or wooden work surface. Make a well in the center and crack in the eggs; add the salt. Gently beat eggs with a fork, incorporating the surrounding flour, until batter is runny. Bring remaining flour into the batter using a bench scraper until dough forms a ball. Mix in water, 1 tablespoon at a time, if dough feels too stiff.
Knead dough with your hands by flattening the ball, stretching it and folding the top towards the center. Turn 45 degrees and repeat until dough is soft and smooth, about 10 minutes.
Shape dough into a ball. Place in a bowl and cover with plastic wrap. Refrigerate until firm, 30 minutes to 1 hour.
Dust work surface with flour. Divide dough into 3 equal portions. Flatten 1 piece a little and roll out 5 to 6 times with a rolling pin. Turn dough 45 degrees and roll 5 to 6 times. Continue rolling and turning until dough is 1/16-inch thick.
Cut dough into a sheet. Dust generously with flour and roll into a cylinder. Slice cylinder into 1/4-inch thick strips. Unroll strips and toss generously in flour. Let tagliatelle air-dry, about 30 minutes.
Bring a large pan of salted water to a boil. Drop in tagliatelle and cook until tender yet still firm to the bite, 3 to 5 minutes. Drain.