4 pounds mixed apples, such as Granny Smith and Cripps Pink, peeled and cut into eighths to make 3 qts.
1 large egg yolk
1 tablespoon heavy whipping cream
How to Make It
Make pastry and chill.
Meanwhile, melt butter with 1 cup sugar and 1/2 tsp. salt in a heavy 5- to 6-qt. pot over medium heat. Cook, stirring constantly, until a smoky haze comes from mixture and it turns deep golden brown, 6 to 9 minutes. Carefully stir apples into caramel (it will bubble and seize up). Cook, stirring often and reducing heat if needed to maintain a steady simmer, until apples look mostly translucent and are tender when pierced with a fork (a few will have fallen apart), about 20 minutes. Transfer apples and caramel to a rimmed baking sheet. Let cool a bit, then chill until cold.
In a small bowl, whisk egg yolk, cream, and a pinch of salt until well blended. Chill egg wash airtight up to 1 week.
Unwrap and lightly flour 1 pastry disk, work surface, and rolling pin. Working from center with short strokes, roll pastry into an even 13-in. round. As you work, lift pastry using a bench scraper or long metal spatula, and reflour board and pin as needed to keep it from sticking.
Fold pastry in half and open up into a buttered regular 9-in. pie pan, easing it into place without stretching. Fill shell with cold apples and caramel; pat into an even mound. Fold overhanging pastry over apples and trim with scissors to a 1-in. border. Chill scraps and pie shell.
Roll out second pastry disk the same way as the first to an 11-in. round. Cut shapes close together with a floured 2 1/2-in. cookie cutter. Transfer to a baking sheet. Gather and reroll all scraps and cut the same way (you'll have about 25 cutouts total). Chill cutouts until firm, 15 minutes.
Brush pie border with egg wash. Lay 10 cutouts in a concentric circle around rim of pie, barely covering the border and barely overlapping one another; brush with egg wash, so they adhere to one another. Make a second circle the same way with 5 cutouts, overlapping the first by about 1/2 in. Set 1 cutout in the center.
Freeze pie and remaining cutouts uncovered until cutouts feel very firm, 20 to 30 minutes. Meanwhile, preheat oven to 350° with racks in center and lower third. Brush pie with egg wash again and sprinkle with remaining 1 tbsp. sugar.
Bake cutouts on baking sheet until well browned, 20 to 22 minutes. Bake pie until crust is deeply browned, juices are bubbling, and an instant-read thermometer inserted in center reaches 180°, 1 1/2 hours; after 45 minutes, set a drip pan on rack under pie, and after about 1 1/4 hours, tent pie with foil. Cool pie on a rack at least 2 1/2 hours. Serve with extra cutouts for nibbling.