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.