Combine the flour, salt, and sugar in the bowl of a food processor. Pulse twice to combine.
Add the grated Parmesan cheese and butter. Pulse until the pieces of butter are the size of peas, about five 3-second pulses.
With the processor running, slowly add 3 tablespoons of water. When a dough is formed and tested by making a fist, it’s done. If it does not hold together, add another tablespoon of water.
Dump the dough onto clear plastic wrap. Knead with the plastic and wrap tightly into a disk. Chill for at least half an hour.
Once chilled, remove from the plastic and roll on a floured surface to a 12-inch rough circle. Place in a 9-inch tart pan or a 10-inch pie dish. Let the dough fall into the corners and remove any excess over the edges. Place shell in the freezer to firm up for 20 minutes.
Once cold, remove from freezer. Line with parchment and weights (or dry beans). Bake at 375°F until the bottom is set, 15 minutes.
Remove parchment and weights and continue baking until crisp and golden, 5-7 minutes.
Remove the shell from the oven and set aside.
For the filling
Melt the butter or bacon grease in a 12-inch skillet.
Add the onion and sauté until translucent, 5 minutes.
Add the sliced mushrooms, salt, and pepper. Continue to sauté until the mushrooms are cooked down, 8 minutes. Set aside to cool.
In a large bowl, combine the eggs, milk and cream. Whisk to combine and break up the eggs.
Add salt, pepper, and mustard. Whisk to combine.
In the prepared shell, evenly add the prepared bacon, cheese, and onion and mushroom mixture.
Pour the egg mixture over the top.
Add some grated Parmesan cheese and bake until the quiche is puffed and golden, 25 minutes.
Cool for 10 minutes before removing from the tart pan and cutting. Slice and enjoy.