4–5 large carrots, peeled and trimmed
3 large eggs
2-1/2 cups all-purpose flour plus more for work surface
1 teaspoon kosher salt
Bring water to a boil in a medium saucepan, add the carrots, reduce the heat and simmer until tender, about 5-7 minutes. Drain the carrots and puree them in a food processor. Measure 1/2 cup carrot puree and discard the rest. Mix the eggs and carrot puree in the food processor until fully combined. Add the flour and remaining salt and pulse until a dough forms.
Turn the dough onto a lightly floured work surface and knead for a couple minutes until the dough is smooth. Form the dough into a round disk and cover with plastic wrap. Allow the dough to rest at room temperature for 30 minutes or refrigerate overnight. (If the dough is refrigerated, make sure to let it come to room temperature before rolling out the pasta.)
Divide the pasta dough into 4 pieces keeping them covered. Working with one piece of dough at a time, flatten slightly and feed through the smooth rollers of a pasta machine starting at the widest setting. Fold the pasta sheet in half each time it is fed through at a narrower setting until the pasta sheet is very thin but does not tear. Place the pasta sheets on a floured work surface to dry for 10 minutes. (Note: If you do not have a pasta machine, you can roll out the sheets of pasta with a rolling pin on a floured work surface.)
Cut the pasta sheets into 12-inch lengths. Sprinkle the sheets with flour and roll them up loosely. Using a knife or pizza cutter, trim the rough edges and cut into 1/4-inch wide strips. Unroll and place the strips on a floured sheet pan. Repeat with the remaining dough.
Bring a large saucepan of salted water to a boil. Cook the pasta for 2 minutes until al denté and then toss with desired sauce.