Clean your countertop well, then mound the flour and make a well in the center. Add the eggs and salt into the well.
Whisk the eggs and salt together within the well, keeping the flour walls intact.
Slowly mix the flour into the eggs, working from the inside edges outward, until a shaggy dough starts to form.
Finish bringing the dough together with your hands, adding water and half-and-half (or heavy cream) a tablespoon at a time only if the dough feels too dry.
Knead until smooth, then wrap tightly in plastic or a sealed bag with the air pressed out. Rest at room temperature for at least 30 minutes, or refrigerate and bring back to room temperature before rolling.
Divide the dough into quarters. Flatten one quarter into a rough rectangle and pass it through the pasta machine on the largest (thickest) setting.
Fold the two long edges of the sheet inward so they meet in the middle, then fold the whole thing in half again, like closing a book. Press down firmly so it holds together.
Feed the narrow, folded edge through the machine again on the largest setting. Repeat this fold-and-roll step once more if the sheet still looks uneven.
Once the sheet is smooth and even, run it through progressively thinner settings until you reach your desired thickness, then cut into your pasta shape of choice.