We like to think of this dessert as the turducken of pies -- spiced, cranberry-stuffed apples are baked in a pie shell, surrounded by cake batter. It's inspired by the bird's-nest pudding in Laura Ingalls Wilder's classic "Farmer Boy."
Preheat oven to 375 degrees. On a lightly floured work surface, roll out dough 1/8 inch thick. Cut out a 13-inch round. Fit into bottom and up sides of a 9-inch springform pan. Freeze 15 minutes.
Blind-bake crust: Line chilled crust with parchment and then foil, pressing into corners to hold up sides. Fill with baking beans or rice. Bake until sides are set and beginning to turn golden, 45 minutes to 1 hour. Remove paper and beans from crust and continue baking until bottom crust is golden brown and dry, about 10 minutes more.
Meanwhile, stir together 1/2 cup sugar, 1/2 cup cranberries, cinnamon, nutmeg, and 1/4 teaspoon salt. Peel apples; scoop out cores from stem ends with a melon baller or a sharp-tipped spoon, leaving bottom ends intact. Fill hollows with 1 heaping tablespoon cranberry mixture, pressing down to fit as much in as possible and mounding slightly over tops. Transfer to an 8-by-8-inch baking dish. Dot with room-temperature butter. Thirty-five minutes before crust is done, place in oven and bake 25 minutes, turning apples halfway through so sides facing edge of dish face in. Cover with foil and bake until apples are just barely tender when pierced with the tip of a knife, 5 to 10 minutes more. Toss remaining 1/4 cup cranberries with juices in bottom of dish.
Meanwhile, whisk together flour, baking powder, and remaining 1/4 teaspoon salt. In a separate bowl, whisk egg yolk with remaining 1/2 cup sugar until combined. Whisk in flour mixture, then melted butter and milk. Beat egg whites until stiff peaks form, then gently fold into batter.
Pour batter into warm crust. Nestle apples into batter, spooning cranberries and 2 tablespoons of juices from dish over top. Return to oven; bake until pie is puffed and beginning to turn golden on edges, and springs back when pressed in center, 30 to 40 minutes. Let cool at least 45 minutes. Serve warm, with ice cream.
The piecrust and the apples should be baked at the same time, side by side or on two separate racks. The entire pie can be made a day ahead and reheated in a 375 degrees oven for 10 to 15 minutes.