Preheat the oven to 425°F.
Place the pie crust into a 9-inch pie dish. Crimp the edges and dock the bottom with a fork. Set aside
Heat the butter in a skillet over medium-low heat. Add the onions with a pinch of salt and cook for 30–40 minutes, stirring occasionally, until deeply golden and caramelized. Let cool slightly.
Shred the zucchini and squeeze out as much moisture as possible using a clean kitchen towel.
In a large bowl, whisk together the eggs, egg yolks, heavy cream, garlic powder, parsley, salt, and pepper.
Fold in the zucchini, caramelized onions, mozzarella, and Parmesan.
Pour the filling into the prepared crust.
Bake at 425°F for 20 minutes, then reduce the oven temperature to 350°F and bake another 30 minutes, or until the center is just set.
If the crust browns too quickly, loosely cover the edges with foil.
Let the quiche cool for 10–15 minutes before slicing.
Garnish with fresh basil and additional Parmesan, if desired.