1. In a 5- to 6-quart pan, bring about 3 quarts water to a boil over high heat. Snap off and discard tough ends of asparagus. Add asparagus to boiling water. Cook, uncovered, until just barely tender to bite, 4 to 5 minutes. With tongs, lift out asparagus and drain; cool slightly.
2. In a blender or food processor, combine lemon juice, oil, garlic, and basil. Return water in pan to a boil. Add fettuccine. Cook, uncovered, until barely tender to bite, about 3 minutes.
3. Meanwhile, diagonally trim off top 3 to 4 inches of asparagus spears. Place asparagus tips in a colander. Coarsely chop stems and add to blender along with 6 tablespoons hot water from pasta pan. Whirl until smooth.
4. Drain pasta into colander containing asparagus tips. Transfer tips and pasta to serving bowl. Pour asparagus purée over pasta and mix. Add salt, pepper, and cheese to taste.