Preheat oven to 375 degrees
Make the Pate Brisee by combining the flour, salt, and sugar in a food processor.
Add the cut-up butter and pulse a couple of times.
Slowly add the ice water a little at a time until the mixture looks like oatmeal. Still separate but if you pinch some in your hand and gently squeeze, it sticks together.
Remove dough from processor and form into a ball. Wrap in plastic and place in the refrigerator for 1 hour.
Meanwhile, saute the onion in 2 tablespoons olive oil on medium heat for about 5 minutes until they start to carmelize.
Add the mushrooms, salt, pepper, and stir. Continue cooking until most of the water has evaporated from the mushrooms. Approximately 10 minutes. Set aside.
After 1 hour remove the dough and roll out into a rough circle about 1/4-1/8 inch thick.
Brush the dough with the remaining 2 tablespoons of olive oil in the center of the dough, leaving about a 2 inch border around the edge. Sprinkle the Parmesan cheese on top.
Next place the mushroom mixture on top of the Parmesan cheese.
Top with sliced mozzarella.
Then fold over the edges of the pate brisee and brush with an egg wash.
Bake for 30-40 until the crust is golden brown.