Add flours and salt to a food processor. Pulse 2 to 3 times to mix. Add butter and pulse 7 to 9 times until there are pea-sized butter pieces. Turn processor on and slowly drizzle ice water until dough slightly comes together. It should be a little crumbly but stick together when pressed.

Place dough on plastic wrap and form a square about 1-inch thick. Cover and refrigerate for 1 hour.

Preheat oven to 350°. Between 2 pieces of parchment, roll the dough out evenly into a 12-inch square. Cut into 4 even square pieces. Spread sauce onto all pieces of dough. Top one side of the squares with cheese, corned beef, and sauerkraut. Fold the dough and cover the filling. Crimp edges with a fork. Place pies on a parchment-lined rimmed baking sheet. Brush the crust with the egg wash and cut tiny vent holes.

Bake 25 to 30 minutes, until the crust is lightly browned. Let it rest 5 minutes on a baking sheet. Serve.

For extra flavor, sprinkle caraway seeds on the hand pies after the egg wash is applied.