Put the flour and salt in a food processor. Pulse a few times to blend. Add the butter and pulse until the mixture resembles cornmeal. Add the ice water 1 Tbs. at a time and process until the dough comes together in a rough ball. Remove the dough, shape it into a disk, wrap it in plastic, and chill for 1 hour while you make the filling and topping.
In a small bowl, combine the goat cheese, mascarpone, garlic, chives, salt, and pepper, mixing with a fork until well blended. Set aside.
In a large skillet, heat the butter and oil over high heat until the butter is foaming. Add the mushrooms and shallots and cook, stirring, until the mushrooms are lightly browned. Add the thyme, salt, and pepper and cook another 1 minute. If using wine, add it as well and cook until the liquid has evaporated, about 4 minutes. Set aside.
Heat the oven to 450°F. Line a baking sheet with parchment. On a lightly floured surface, roll out the chilled dough into a 13-inch round. Transfer the dough to the parchment-lined sheet. Spread the cheese mixture to within 1 inch of the edge. Top with the mushrooms. Fold the edge of the dough over the filling, pleating it as you go. Bake until golden brown, about 30 minutes. Serve warm.