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. Purée carrots in a food processor (you should have about 1/2 cup purée).
Add eggs and yolk to purée in food processor, and process until combined. Add flour and 1 heaping teaspoons salt, and process until dough just comes together about 20 seconds.
Transfer dough to a well-floured surface. Knead until smooth and elastic 5-10 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 1-2 hours.
Cut dough into 8 pieces. Working with 1 piece at a time (keep the remaining piece covered with the inverted bowl), flatten dough into an oblong shape slightly thinner than the pasta machine's #1 setting. 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.
Turn the dial to the next narrower setting. Pass dough gently twice, gently supporting it with your palm. Continue to press dough, even finer settings, two passes on each setting, until sheet is almost translucent and very thin but still intact (#5of 8 on a kitchen Aid 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.
FOR FARFALLE, LASAGNE, SQUARES, OR RAVIOLI:
Place rolled sheets on lightly floured surface and cut out as directed in corresponding recipe, using a dry brush to dust flour off if needed.
FOR RIBBON PASTA:
Place dough on a drying rack until slightly tacky, 10-15 minutes, and then use a pasta machine or cutting attachment to cut into strands. Drape over rack until strands are semi-dry and won't stick together, about 20 minutes. Cook immediately, or lay flat on an baking sheet dusted with semolina, cover tightly with plastic wrap, and refrigerate overnight,.
FRESH SPINACH DOUGH VARIATION:
Add 2-inches of water to a medium saucepan, and fit with a steamer insert. Bring to a simmer. Add 6oz. spinach, cover and steam until bright green and softened, about 2 minutes. Let cool slightly. Squeeze out liquid using a clean kitchen towel or paper towels. Purée spinach in a food processor (you should have about 1/2 cup of purée). Proceed with instructions for fresh carrot pasta dough starting with step 2.
FRESH RED PEPPER PASTA DOUGH VARIATION:
Preheat oven to 450°F Rub 1 large red bell pepper with 1/2 tsp olive oil, and place on a baking sheet. Cook, flipping after 10 minutes, cooking until skin begins to bubble and is lightly brown in spots, about 25 minutes. Transfer to bowl, and immediately cover with plastic wrap. Set aside to steam for 15 minutes. Remove skin, stem, seeds and ribs, and discard. Purée pepper in a food processor (you should have about 1/2 cup purée). Proceed with instructions for fresh carrot pasta dough starting with step 2.
FRESH BEET PASTA DOUGH:
Preheat oven to 375°F Toss 2 medium beets, without greens, with 2 teaspoons oil and a large pinch of salt. Wrap tightly in a parchment-lined piece of foil, and place on a baking sheet. Roast until tender, 45 minutes to 1 hour. Let cool. Rub with paper towels to remove skins. Purée in food processor (you should have 1/2 cup purée). Proceed with instructions for fresh carrot pasta dough starting with step 2.