In a food processor, whirl flour and butter until mixture looks like cornmeal. Add 1 egg and mix just until dough holds together. Press evenly over bottom and 1¼ inch up the sides of a 9-inch springform pan; chill.
Preheat oven to 375 degrees.
Blanch asparagus in a saucepan of boiling water until barely tender-crisp, about 45 seconds. Drain, transfer to a bowl of ice water, drain again, and pat dry.
In a bowl, whisk remaining 3 eggs to blend. Whisk in half-and-half, salt, and pepper. Sprinkle gruyere and green onion in prepared crust, then arrange asparagus and morels on top. Pour egg mixture slowly over vegetables.
Bake quiche on bottom rack until filling no longer jiggles when gently shaken, 40 to 45 minutes. Let cool in pan on a rack at least 30 minutes. Loosen quiche from pan rim with a knife, remove rim, and slice. Serve warm or at room temperature.