In a large and shallow mixing bowl, add the flour and create a well in the middle with your hand.
In a small bowl, whisk together the egg and water, and pour into the flour well.
Using a fork or your fingers, slowly incorporate the egg with the flour until a shaggy crumble appears. You can continue bringing the dough together in the bowl or pour it out onto a clean work surface. It will take about a minute to get the dough to really come together. If some flour won't incorporate, toss it.
Knead the shaggy dough for 6-8 minutes until it becomes soft and smooth. Wrap tightly in plastic wrap and rest at room temperature for at least an hour or in the refrigerator for one night if you aren't using it right away. Just make sure you bring it to room temperature when you are ready to use it.
Work with the dough in two parts. Cut the dough into two equal sections using a bench knife and wrap the reserved piece of dough back in the plastic wrap to keep it from drying out.
With your hands, press the working dough into the rough shape of a rectangle and feed it through your pasta rolling machine at the thickest level. Fold the rolled dough over onto itself and feed it through the roller again. You can do this a few more times if the dough needs a little extra strengthening or shaping.
Change the thickness of the roller to the next thinnest option and continue feeding your pasta dough through the machine at a thinner and thinner setting until you reach the thinnest possible setting. The dough will be long and nearly paper-thin.
Use the backs of your hands to transport the dough to a floured work surface. Cut your pasta shapes and repeat for the other portion of dough.
Leave the cut pasta on the counter to dry before cooking so that it won't stick together in the pot. Drying time will depend on the shape and size of the pasta. If you are storing the pasta, 24 hours is best.
When you are ready to cook, bring a pot of well-salted water to a boil, reduce the heat, and add your pasta in batches so as not to overcrowd the pot. Fresh pasta can cook quickly in just a few minutes, so keep an eye on it and test the chew frequently. If you are making ravioli, wait for the ravioli to start floating at the top and cook for an additional few minutes.
