Heat the oil and butter in a large skillet over medium heat. Add garlic, and cook until golden, about 5 minutes. Stir in shallot, and cook until translucent, about 3 minutes. Add bell peppers and red pepper flakes, and season with salt and pepper; cook, stirring until soft, about 5 minutes more. Stir in 1/2 of the basil.
Meanwhile, bring a large pot of salted water to a boil. Add pasta, and cook until al dente, 2 to 4 minutes. Drain, and add to the pan with peppers. Cook, stirring constantly, for 1 minute. Transfer to serving platter. If desired, remove garlic, and discard. Garnish with remaining basil and Parmesan. Serve immediately.
Preheat oven to 450 degrees F. Rub outside of pepper with olive oil, and place on a small baking sheet. Cook until skin begins to bubble and turn lightly brown in spots, 20 to 25 minutes, turning once after 10 minutes. Transfer to a container with a tight-fitting lid to steam for about 15 minutes. Remove stem and seeds; peel away skin, and discard.
In the bowl of a food processor, puree pepper (there should be a generous 1/2 cup puree). Add flour and eggs. Process until dough forms a ball, about 15 seconds. Transfer dough to a lightly floured work surface. Knead until smooth and elastic, about 5 minutes. Cover with an inverted bowl or plastic wrap, and let rest 1 1/2 hours at room temperature or refrigerate overnight.
Divide dough into 8 pieces. Working with a few pieces at a time and keeping remaining pieces covered, flatten each portion of dough into a disk shape somewhat narrower than the machine's opening. Very lightly dust dough with flour. Feed through at machine's widest setting. (If dough pulls or tears when passing through machine, simply sprinkle a little more flour over dough, just before it's fed, to keep it from sticking. When finished, remove excess flour with a dry pastry brush.) As pasta sheet emerges, gently support it with palm, and guide it onto work surface. Fold sheet in half. Pass through a second time on the same setting to smooth dough and increase its elasticity.
Thin dough by passing it through ever-finer settings, 2 passes on each setting, until sheets are almost translucent but do not tear. Pass the dough dthrdough until it has been passed through settings 1 through 4 (machine settings differ?some have as many as 10, while others have only 6).
Switch to the cutter attachment, and pass each sheet through fettuccini cutter. Separate strands, and hang on a pasta rack to dry so that they do not stick together.
Yield: about 1 1/2 pounds
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