1. Heat oven to 450 degrees. Place three-quarters of the pate brisee on a lightly floured work surface. Roll dough out to a 12-inch-diameter circle, 1/8 inch thick, dusting work surface with flour to prevent sticking, as necessary. Brush off excess flour; roll dough around rolling pin, and lift it over a deep-dish pie pan. Line the pan with the dough, pressing it into the corners. Trim dough so that it hangs over pie plate by about 1/4 inch. Roll out remaining dough to a 10-inch-diameter circle. Transfer dough round to a parchment-lined baking sheet, and refrigerate both crusts.

2. Peel and core apples, then cut into 1/4-inch-thick slices. As you cut, place slices in a large bowl, and sprinkle with lemon juice to prevent discoloration. In a small bowl, combine sugar and cinnamon. Toss apple slices with sugar mixture.

3. Divide butter between 2 large skillets, and melt over medium-high heat. Divide apples between skillets, and cook, stirring often, until sugar melts and apples are golden and coated in syrup but still firm, 8 to 10 minutes.

4. Pour brandy into a measuring cup. Pour half the brandy into 1 skillet and carefully ignite with a match. Cook until flames die down, shaking pan to toss apples in melted sugar and brandy. Repeat with remaining brandy and apples. Remove from heat, and set aside.

5. Remove the crusts from the refrigerator; transfer cooked apples to prepared pie pan, letting apples mound in center. Whisk egg yolk with 2 tablespoons water to make egg glaze, and brush glaze on edge of dough. Center the rolled pie dough over apples. Tuck edges of top crust between pie pan and bottom crust. Using your fingers, gently press crusts together along edge, and crimp.

6. Cut several steam vents into crust. Brush surface with egg glaze, and sprinkle with remaining 1 tablespoon sugar.

7. Bake pie until golden brown on top, about 15 minutes. Reduce heat to 350 degrees and bake until the crust is crisp and the apples are soft, 45 to 50 minutes more. Let stand at least 30 minutes before serving.