Preheat oven to 425 degrees F. Let piecrust stand at room temperature according to package directions. Line a 9-inch pie plate with piecrust. Crimp edge as desired. Line unpricked pastry with a double thickness of foil. Bake for 8 minutes. Remove foil. Bake for 4 to 5 minutes more or until pastry is set and dry. Remove from oven. Reduce oven temperature to 375 degrees F.
Meanwhile, trim top off fennel bulb, reserving some of the feathery leaves for garnish. Trim fennel bulb and thinly slice. In a covered medium saucepan, cook fennel, asparagus, and onion in a small amount of boiling water for 4 to 6 minutes or just until vegetables are tender. Drain vegetables; set aside.
In a medium bowl, whisk together milk and flour until smooth. Add eggs, snipped or dried basil, salt, and pepper, whisking until combined.
Spoon fennel mixture into baked piecrust. Sprinkle with cheese. Slowly pour egg mixture over all.
Bake, uncovered, for 30 to 35 minutes or until egg mixture is set in center. If necessary to prevent overbrowning, cover edge of pie with foil for the last 5 to 10 minutes of baking. Let stand for 10 minutes before serving. Top with reserved fennel leaves and, if desired, fresh basil leaves.