On a work surface, toss the flour with the minced sage. Make a well in the center and pour in the egg mixture. Stir with a fork, gradually incorporating all of the flour. Once the dough is too stiff to stir, knead it to work in as much of the flour as possible. Divide the dough in half, wrap in plastic and let rest at room temperature for 30 minutes.
Lightly flour the dough and, working with one piece at a time, run it through a pasta machine set on successively narrower notches, ending at the second-to-thinnest setting. Cut the pasta sheet into 4 pieces and drape over a rack to dry for a few minutes. Repeat with the remaining dough. Run the pasta sheets through the fettuccine cutter, lightly dusting with flour. Spread the fettuccine out on a baking sheet and let stand for 15 minutes.
Cook the fettuccine in boiling salted water until al dente, about 5 minutes. Drain well, reserving 3/4 cup of the cooking water. Return the water to the pot and stir in the butter, olive oil and a pinch of salt and pepper. Add the fettuccine and cook over moderate heat for 2 minutes. Add the cheese, toss and serve.
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