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. 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. Refrigerate dough for 2 hours.
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.
Allow pasta to dry for one hour prior to cooking.
Cook in a large pot of boiling water until al dente. This should take only 3 to 5 minutes depending on the thickness of the pasta.