Prepare the Crust: Add flour, powdered sugar, and salt to a food processor, and pulse until combined, 2 to 3 times. Add butter; process until mixture resembles coarse sand.

Whisk together egg yolks and ice water in a small bowl. With processor running, pour egg mixture through food chute, and process until mixture begins to clump.

Transfer to a lightly floured surface; knead until dough is smooth, 2 to 3 times. Shape dough into a 7- x 5-inch rectangle, and cover with plastic wrap. Chill 1 hour.

Prepare the Filling: Preheat oven to 350°F. Combine all berries in a large bowl. Stir together sugar, tapioca, nutmeg, and salt in a small bowl. Add to berries; toss gently to coat. Scoop berry mixture into a 14- x 10- x 3-inch baking dish (or a 13 ½- x 9 ⅝- x 2 ¾-inch disposable aluminum pan). Dot top of berries evenly with butter.

Roll dough on a lightly floured surface into a 16- x 12-inch rectangle. Use a ¾-inch round cutter or base of a piping tip to cut holes all over dough, leaving a 2-inch border around edges. Place dough on top of berry mixture; fold dough under edges of pan and crimp. Whisk together egg yolk and water; brush dough evenly with egg mixture.

Bake in preheated oven until Filling is bubbly and Crust is well browned, 1 hour and 10 minutes to 1 hour and 15 minutes. (Cover top with aluminum foil, if needed, to prevent overbrowning.) Let stand on a wire rack at least 2 hours before serving.

Prepare the Dip: Combine half-and-half, sugar, and salt in a saucepan over medium; bring to a gentle boil. Stir together cornstarch and cold water in a small bowl until smooth, and stir into half-and-half mixture. Cook until mixture is thick and bubbly, stirring constantly, 1 to 2 minutes. Strain Dip through a fine wire-mesh strainer into a small serving pitcher, and stir in vanilla extract. Serve sonker warm, and drizzle each serving with warm Dip.