Place basil in a food processor; cover and process until finely chopped. Add flour; process until blended. Add the eggs, water and oil. Process for 15-20 seconds or until dough forms a ball.
Turn onto a floured surface; knead until smooth and elastic, about 8-10 minutes. Cover and let rest for 30 minutes. Divide into fourths.
On a floured surface, roll each portion of dough to 1/16-in. thickness. Roll up jelly-roll style and cut into 1/4-in. slices. Separate and unroll the slices. Hang to dry or let stand on a clean towel for 1 hour. Cook noodles in boiling salted water for 4-5 minutes or until tender; drain.