While the Pie crust is par baking, place grated zucchini in a colander or sieve, sprinkle with salt, about ½ teaspoon, tossing the zucchini so the salt is mixed in. Let the zucchini soak for at least 5 minutes and up to whenever you're ready to add it to the pie crust to release some of the moisture. While the zucchini is resting, prepare the other ingredients if you haven't already; cut the corn from the cob and grate the Monterey Jack cheese.
Place the cream cheese in the bowl of a stand mixer, or medium bowl if using hand beaters. Beat on medium until smooth. Add the cream, egg yolks, and egg, beating between each addition. Add ½ teaspoon salt and black pepper to taste. Scrape down the bowl and beat again on medium for about 30 seconds. If you still have tiny pieces of cream cheese that won't smooth, no worries.
Squeeze the grated zucchini with your hands to remove excess moisture. You can also place the zucchini on a triple layer of paper towel and roll it up tightly to remove moisture (you can do both!). Set aside.
Place the cooled pie crust on a baking sheet. Sprinkle one cup of Monterey Jack cheese over the bottom of the pie crust (see notes below if there are any holes in your baked crust), then add the corn, distributing evenly. Now add the zucchini, separating and fluffing with your fingers. Slowly pour the cream cheese/egg mixture over the zucchini. This will make the crust very full, but not to worry, that's expected. Sprinkle the last one cup of Monterey Jack cheese over the quiche. Grind some extra black pepper over the top of the cheese. You can add some very thin slices of zucchini around the top for a pretty presentation and a few extra kernels of corn, if you wish. If you want the pie crust edges to shine, brush a tiny bit of the cream cheese/egg mixture on the top, either with a pastry brush or your fingers.
Carefully place the baking dish with the quiche in the oven (the oven should be at the new temperature of 350F°) and bake for 45 minutes. After about 30 minutes, start checking the quiche and if the crust or top seems to be getting browned too quickly, place some aluminum foil over the quiche to slow the browning.
After 45 minutes, check the quiche for doneness by jiggling the baking sheet and if the filling is puffed and golden, with just a little movement in the center, insert a knife in the center. It should come out pretty clean except for the melted cheese. If you did cover your quiche with aluminum foil, it may not be quite done. If more time is needed, return to the oven for another 5 minutes. Even if the top of the quiche becomes very brown, it's still delicious.
Remove the quiche to the stove top and let cool until the top evens out a bit, about 5 minutes. Serve immediately, or cool to room temperature and refrigerate to serve later.