On a lightly floured surface, roll dough into a 12-inch circle, about 1/8 inch thick. Fit into a 10-by-1 1/2-inch fluted round tart pan with a removable bottom, pressing dough into corners. Trim dough flush to edges. Transfer to refrigerator to chill for 30 minutes.
Preheat oven to 375 degrees. Line pastry with parchment paper-lined aluminum foil, pressing into the corners and edges. Weight with beans, rice, or aluminum pie weights. Bake until golden brown and dry, 40 to 50 minutes. Carefully remove foil and weights. Transfer to a wire rack to cool while making filling.
Heat oil in a large nonstick skillet over high heat. Add shallots, and cook, stirring, until translucent but not brown, about 1 minute. Add mushrooms, and season with salt and pepper. Cook, stirring frequently, until mushrooms first release their liquid and then liquid evaporates and mushrooms are dark golden brown, 8 to 10 minutes.
Transfer tart pan to a baking sheet. Sprinkle half the cheese evenly over the bottom of the crust. Sprinkle with mushrooms, then top with remaining cheese. In a medium bowl, whisk together milk, cream, eggs, and egg yolk. Season with nutmeg, salt, and pepper. Pour over cheese. Transfer to oven, and bake until just set in the center, 30 to 35 minutes. Cool on a wire rack for about 10 minutes before slicing.