Preheat oven to 375 degrees. Toss beets with 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 beets with paper towels to remove skins. Puree in a food processor (you should have 1/2 cup puree).
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.
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 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.
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.
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.
For farfalle, lasagna squares, or ravioli: Place rolled sheet on a lightly floured surface, and cut into desired shape, using a dry brush to dust flour off if needed. Cook according to corresponding recipe. For ribbon pastas: Place dough on a drying rack until slightly tacky, 10 to 15 minutes, and then cut into strands using a pasta machine, cutting attachment, or by hand. Drape over rack until strands are semidry and don't stick together, about 20 minutes. Cook immediately, or lay in a single layer on a baking sheet dusted with semolina, cover tightly with plastic wrap, and refrigerate overnight.