In a large bowl or on a clean work surface make a mound out of the flour then make a deep well in the middle. Crack the eggs into the well of flour. Add a generous sprinkle of salt. Using a fork, gently beat the eggs until combined then start gradually scraping the flour into the egg mixture until the mixture thickens and a dough starts to form. At this point use your hands to continue adding flour to the egg mixture just until the dough comes together in a cohesive ball. You may not need to use all the flour. Continue gently kneading the dough, adding more flour if it is too sticky, until it becomes smooth, soft, and elastic.
Cover the dough and let it rest at room temperature for about 15 to 20 minutes or up to 2 hours. The dough can also be completely covered and refrigerated for up to 1 day at this point (the dough may turn a grayish color from refrigeration which should not impact the taste).
Prepare your pasta roller and generously flour your work surface. Cut off a quarter of the dough and coat completely in flour. Flatten the dough slightly before rolling through the widest setting on the pasta maker. If the dough comes out oddly shaped, fold the edges over to reform into a perfect rectangle. Continue to run the dough through the widest setting and reshaping until the dough is of even thickness and shape and has increased elasticity. If you're using a hand crank pasta roller, try to maintain a consistent rolling speed. Continue rolling the dough through each roller setting twice to ensure the dough remains evenly thick, well shaped, and satiny in texture. Roll until desired thickness.
Use the pasta roller cutter attachments to cut the dough into desired pasta shapes. If the dough isn't completely cut but the cutter attachment, it may be too soft and needs more flour. You can also flour the dough well, roll it up, and use a knife to cut strips of dough. The pasta is now ready to be cooked or may be covered and refrigerated for up to 2 days. To dry pasta cuts like fettuccine, either dry the pasta on a pasta drying rack (or a makeshift one) or curl the pasta into nests and let the pasta dry for 24 hours at room temperature. Filled pasta can be frozen in a single layer on floured baking sheets then transferred to airtight containers for up to 2 months. No need to thaw before cooking
Variations:
Traditional: Replace all-purpose flour with Italian "00" flour.
Whole Wheat Pasta Dough: Use 1 1/4 cups all-purpose flour and 1 1/4 cups whole wheat flour and proceed with recipe.
Herbed Pasta Dough: Add 1/4 cup minced fresh herbs to the eggs, adjusting flour if necessary.
Spinach Pasta Dough: Cook 6 ounces of spinach in 1/4 cup water in a large pot over medium heat until the spinach is wilted and tender, about 4 to 5 mintues. Drain the spinach in a dry cheesecloth and squeeze out as much water as possible. Puree the spinach in a blender or food processor until smooth. Proceed with dough, decreasing the eggs to 3 and increasing the flour if necessary.