Remove the dough from the plastic wrap and plate it on a floured surface. Roll it out into a circle of 14 to 15 inches in diameter. It helps to flip over the dough and rotating it in quarter turns, while lightly flouring your surface, in order to get an even circle.
Carefully transfer the rolled dough to the lined baking tray, lining up the center of the dough with the center of the baking tray. The sides of the dough might go over the tray, but they will be folded in later.
Spread the cheese filling evenly in the center of the dough, leaving a 3 inch border of dough all around.
Place the mushroom filling on top of the cheese, making sure to stay on the same surface area as the cheese filling.
Fold over a 4-inch piece of the border dough over the filling. Continue doing so, a section at a time, overlapping on the last section, until all sides are folded in.
Mix together the egg with a tablespoon of water or milk to make an egg wash. Brush some of the egg wash on the crust. Sprinkle the crust with the Everything Bagel Seasoning.
Bake the galette in the preheated oven for 15 minutes. Turn the tray around and keep baking for another 10 minutes, or until the dough golden brown.
Remove from the oven and let rest 5-10 minutes before cutting into it.