Step 1

Add 2 inches of water to a medium saucepan, and fit with a steamer insert. Bring to a simmer. Add carrots; cover, and steam until softened, about 10 minutes. Puree carrots in a food processor (you should have about 1/2 cup puree).

Step 2

Add eggs and yolk to puree in food processor, and process until combined. Add flour and 1 heaping teaspoon salt, and process until dough just comes together, about 20 seconds.

Step 3

Transfer dough to a well-floured surface. Knead until smooth and elastic, 5 to 10 minutes, adding up to 2 tablespoons flour if dough is too sticky. Place on a piece of parchment, and cover with an inverted bowl, or wrap tightly in plastic; let rest for 1 to 2 hours.

Step 4

Cut dough into 8 pieces. Working with 1 piece at a time (keep the remaining pieces covered with the inverted bowl), flatten dough into an oblong shape slightly thinner than the pasta machine's widest setting (number 1). Dust dough very lightly with flour, and feed through machine. Fold lengthwise into thirds and rotate 90 degrees. Repeat twice on same setting to smooth dough and increase its elasticity.

Step 5

Turn the dial to next narrower setting. Pass dough through twice, gently supporting it with your palm. Continue to press dough, passing it through ever-finer settings, two passes on each setting, until sheet is almost translucent and very thin but still intact (number 5 of 8 on a KitchenAid pasta roller). The dough will stretch to about 16 inches long. If dough bubbles or tears, pass it through again, and dust with flour if the dough is sticking.

Step 6

For farfalle, lasagna squares, or ravioli: Place rolled sheet on a lightly floured surface, and cut out as directed in corresponding recipe, using a dry brush to dust flour off if needed. For ribbon pastas: Place dough on a drying rack until slightly tacky, 10 to 15 minutes, and then use a pasta machine or cutting attachment to cut into strands. Drape over rack until strands are semidry and won't stick together, about 20 minutes. Cook immediately, or lay flat on a baking sheet dusted with semolina, cover tightly with plastic wrap, and refrigerate overnight.