In the bowl of a stand mixer fitted with the paddle attachment or a large bowl, add the flour and salt.
Make a well in the center. Add the eggs and mix until well-combined.
Stir in the wine, followed by just enough water to create a soft dough. If too wet, add a little more flour. If too dry, add a little more water.
On a lightly floured surface, knead the dough until smooth and elastic. Wrap in plastic and allow to rest at room temperature for 30 minutes or up to 1 hour.
Divide the dough into four equal pieces. Place one on a floured work surface and cover the remainder.
Dust a large, rimmed baking sheet with flour or fine semolina.
Roll the dough into a sheet about 1/8th inch (3 millimeters) thick, dusting with flour as needed to prevent sticking. If using a pasta machine, start with the widest setting and gradually roll the sheet to the second to thinnest setting, dusting with flour as needed. The sheet should still have a slight thickness to hold the shape of the stamp on each side.
Use the hollow cutter side of the Corzetti stamp to cut out circles as close together as possible from the thin sheet of pasta.
Dust the Corzetti stamps with flour, continuing to do so as needed to prevent sticking between batches.
Place the larger Corzetti stamp on the work surface with the stamped side up.
Place a cut out circle on the stamp. Cover the circle with the other stamp and press down to create the embossed pattern.
Remove the circle to the prepared baking sheet and repeat with the remaining circles, arranging them in a single layer. Repeat with the remaining dough. Once the baking sheet has been filled with a single layer of embossed circles, cover with a piece of flour-dusted parchment and make another layer.
To cook, bring a large pot of salted water to a boil. Add the pasta in batches, careful not to overcrowd, and boil until they rise to the surface and are just tender, about 3 minutes if fresh.
Uncooked pasta can be frozen in a single layer, then transferred to a freezer-safe bag. Take care, the thin shape can easily snap and break during storage. They can be boiled straight from the freezer, though a minute or two may need to be added to the cooking time.