Using a stand mixer with a dough hook, beat eggs, salt, and water. Gradually add flour until mixture forms a ball.
Knead dough on a floured surface, gradually adding more flour as needed so that the dough is still soft but not sticky.
Transfer to an oiled bowl, cover with a towel, and let rest for at least 1 hour. When dough has rested, cut off a portion while leaving the remaining amount covered.
Press the dough into a rectangle and roll it through a pasta machine at least two times at the widest setting. Reduce setting and process the dough two more times, and then repeat the step. The dough should be about 1/8-inch thick, 12-18 inches long, and 3-4 inches wide. Keep dough dusted with flour as you work.
Lightly dust a work surface and lay out pasta. On half of the sheet, drop tablespoons of your favorite filling about 2 inches apart. Fold the other half of the dough over the filling.
With your fingers, gently press out air pockets around each mound of filling. Use a sharp knife or ravioli cutter to cut out pillowed squares, and crimp the edges of each ravioli to seal. Lightly dust a baking sheet with flour and arrange ravioli in a single layer.
Repeat with remaining dough and filling. In a large pot of boiling salted water, cook batches of the ravioli until they float to the top (about 4 minutes) and remove with a slotted spoon.