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Using a food processor, process basil leaves until chopped very fine. Add 1 1/2 cups of flour and pulse two or three times, or until combined.
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Add egg, 1 teaspoon oil, and the water until dough forms a ball shape. If dough seems dry, add a bit more water. Wrap dough in a piece of plastic wrap which has been coated in a few drops of olive oil.
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Refrigerate dough for 2 hours.
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Remove dough from refrigerator, and cut into 6 equal size portions. Run pasta though pasta machine, or roll with rolling pin to desired thickness. Use the additional flour to coat pasta while rolling.
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Allow pasta to dry for one hour prior to cooking if making noodles.
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Cook in a large pot of boiling water until al dente. This should take only a 3 to 5 minutes, depending on the thickness of the pasta.