Put 2 1/2 cups of the flour into a bowl, make a well in the flour, and add the cheese and eggs. Gradually work the mixture together, adding more flour if necessary, to make a soft but not sticky dough. On a lightly floured surface, knead the dough until it is smooth. Let the dough rest at room temperature, covered with an inverted bowl or wrapped in plastic, for 30 minutes.
Form the dough into a round and cut into quarters. Working with one quarter at a time (cover the remaining dough with an inverted bowl to keep the dough from drying out), on a lightly floured surface, roll the dough into a rope 1/4 inch in diameter. With a knife, cut the rope into 1/2-inch pieces. With your index and third fingers held together, gently press down on each piece, beginning at the top and moving down toward the bottom, dragging your fingers toward you and causing the pasta to roll over on itself. Transfer the formed pasta to a lightly floured jelly-roll pan and let dry at room temperature for at least 30 minutes.
Bring a saucepan of salted water to a boil, and add the pasta. Cook until the cavatelli are al dente, 6 to 8 minutes. They're done when they float up to the top. Drain and toss the desired quantity with your favorite sauce. Freeze leftovers.