Boil stalk pieces, covered, until tender-crisp, about 8 minutes. After 6 minutes, drop in tips. With strainer, lift out asparagus, turn into bowl and cover. Add more water to pot, if necessary, and keep it boiling.
Thinly slice green garlic, using all of green stem. Separate white parts and chop finely. Heat oil in large heavy skillet over medium-high heat. Add white pieces and saute, stirring, until tender, about 2 minutes. Add stems and leaves and saute another minute. Remove from heat.
Boil penne according to package directions until al dente, about 8 to 9 minutes. When pasta is almost cooked, return skillet with garlic in it to medium-high heat. Add asparagus, 1/2 cup parsley, lemon juice and butter. Turn heat to low. Add 2/3 cup pasta water to skillet. Whisk sauce thoroughly to blend.
Drain pasta and turn into serving bowl. Season sauce to taste with salt and pepper. Spoon sauce over pasta and toss. Serve at once in flat bowls. Add sprinkling of cheese and parsley to taste. Makes 4 servings.