Preheat oven to 350 degrees F (175 degrees C).

Place egg white of 1 egg into a small bowl and whisk with water until foamy. Set the egg yolk aside in a large bowl.

Fit pie crust into a 9-inch pie dish. Brush egg white mixture over the edges and inside of crust. Poke holes into the inside and up the sides of the crust with a fork.

Bake crust in the preheated oven until the egg white glaze cooks onto the crust, about 5 minutes.

Pour any remaining egg white mixture into the large bowl with the yolk; whisk in 2 more eggs and milk until thoroughly combined. Set egg mixture aside.

Heat canola oil in a large skillet over medium heat; cook and stir mushrooms and onion until the mushroom's liquid has evaporated, 9 to 15 minutes. Season with salt and black pepper.

Turn off heat and stir spinach into hot mushrooms to wilt. Mix in tarragon, paprika, and cayenne pepper.

Gently stir mushroom-spinach mixture into eggs and milk. Fold 1 1/4 cup of white Cheddar cheese into the filling.

Pour mushroom-spinach filling into the prebaked pie crust and arrange slices of zucchini in a decorative pattern on top of the filling. Top with 1/4 cup white Cheddar cheese in an even layer.

Place the quiche onto a baking sheet and bake in the oven until top is lightly browned and a knife inserted into the middle of the quiche comes out clean, 35 to 40 minutes.

Allow quiche to cool on the hot baking sheet for 5 minutes before removing to finish cooling on a wire rack until serving temperature, 10 to 15 more minutes.