Preheat the oven to 400°F (200°C). Line the pie shell with parchment and fill it with pie weights or dried beans; bake 12–15 minutes until the crust is lightly golden. Remove from oven and let cool to room temperature.
Reduce oven temperature to 350°F (175°C).
In a medium bowl, whisk together the eggs, heavy cream, milk, salt, and pepper until fully combined to make the custard.
Roughly chop the spinach if needed and spread it evenly in the bottom of the pre-baked pie shell.
Layer the diced cooked chicken, diced sun-dried tomatoes, and sliced green onions over the spinach, then sprinkle the mozzarella and Parmesan evenly on top.
Pour the egg custard over the filling, place the pie on a baking sheet, and bake at 350°F (175°C) for 45–55 minutes, until the center is just set and not wobbly.
Remove the quiche and let it rest for 20 minutes before slicing and serving warm or at room temperature.
