Heat oven to 400°F Unfold the pastries and cut each one into 4 squares. Place them on a baking sheet. Using a plastic fork prick the dough all over to avoid puffing.
Meanwhile, heat the oil in a large skillet over medium heat. Add the mushrooms and shallots and season with 1 tsp salt and 1/2 tsp pepper. Cook, tossing occasionally until the mushrooms are browned and tender, 4-5 minutes. Add the wine and cook until nearly evaporated, about 1 minute more.
Dividing evenly, sprinkle the pastry squares with half of the cheese, leaving a 1/2 inch border on each. Top with the mushroom mixture and remaining cheese. Bake until the pastry is golden brown, 20-30 minutes.