Preheat your oven to 375°F (190°C). Place the pastry shell on a baking sheet.
Blanch the asparagus: Bring a small pot of salted water to a boil. Add the trimmed asparagus and cook for 2-3 minutes until bright green and slightly tender. Immediately transfer the asparagus to an ice bath to stop cooking, then drain and pat dry. Cut the spears into 1-inch pieces.
Arrange the cooked asparagus pieces evenly over the bottom of the pastry shell. Sprinkle the Gruyère cheese over the asparagus.
In a medium bowl, whisk together the eggs, heavy cream, milk, salt, pepper, and nutmeg until well combined.
Slowly pour the egg mixture over the cheese and asparagus in the pastry shell.
Sprinkle the Parmesan cheese evenly over the top.
Bake for 40 to 50 minutes, or until the center is set and a knife inserted near the center comes out clean. The top should be golden brown.
Let the quiche cool on a wire rack for at least 15 minutes before slicing and serving.
Notes
You can blanch the asparagus up to one day ahead and store it in the refrigerator.
For a flakier crust, blind bake the pastry shell for 10 minutes before adding the filling.
If you do not have Gruyère, Swiss cheese is a suitable substitute.