►FILLING: Heat oil in a large skillet over medium heat. Add onion and sauté until softened and translucent, about 5 minutes. Add mushrooms, season with salt and pepper, and continue to cook, stirring occasionally, until mushrooms have released their liquid and have become browned and tender, 7 to 9 minutes. Remove from heat, stir in balsamic, and let cool for 10 minutes.
►GALETTE: Arrange a rack in the middle of the oven and heat to 400°F. Line a baking sheet with parchment paper.
►Place the pie crust on a lightly floured work surface. Using a rolling pin, roll the dough out to a round about 12 inches in diameter.
►Starting on one end of the dough, loosely roll up the pie crust around the rolling pin. Transfer it to the prepared baking sheet and unroll the dough back out flat.
►Spread mustard on the dough, leaving about a 1½ to 2-inch border. Sprinkle half the cheese over mustard. Spoon filling evenly on top. Gently fold the edges of the dough over the filling, covering about 1½ to 2 inches of the filling and pleating the dough every 2 inches as you go. Sprinkle top with remaining cheese.
►Bake until the crust is golden-brown, 20 to 30 minutes. Let the galette cool at least 5 to 10 minutes before sprinkling with parsley, cutting into wedges, and serving.