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In a food processor, pulse together flour and salt. Add eggs, yolks, and oil and run the machine until the dough holds together. If the dough looks dry, add another teaspoon olive oil. If the dough looks wet, add a little flour until dough is tacky and elastic.
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Empty dough onto a work surface and knead briefly until very smooth. Wrap in plastic and rest at room temperature for 1 hour or in the fridge overnight.
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Cut the dough into 4 pieces, keeping them covered with plastic wrap or a dish towel when not in use. (If you’re rolling the dough out by hand, rather than using a pasta machine, cut it into 2 pieces instead.) Using a pasta roller set to the thickest (widest) setting, roll one piece of dough out into a sheet. Fold the sheet in thirds like a letter and pass it through the machine 2 more times on the same setting.
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We use a pasta roller and rarely go past the 6 line mark, as the pasta becomes to thin even for angel hair. As you become more familiar with the texture and workability of the dough it will become second nature.
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Place cut pasta on a flour-dusted sheet tray and cover with a dish towel while rolling and cutting the remaining dough. Make sure to sprinkle flour over the cut pasta before you place another layer on top. If not using immediately, cover the sheet pan with a dish towel to keep the dough pliable.
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Bring a large pot of well-salted water to a boil, add fresh pasta and boil for 1 to 3 minutes, depending on the thickness of the pasta. Drain well, but don’t rinse.