Line a rimmed baking sheet with parchment paper; set aside. Remove pie dough from refrigerator. If using homemade dough, let stand at room temperature until slightly softened, 5 to 10 minutes.

Generously flour work surface and use a rolling pin to roll 1 dough rectangle to a 16-by-11-inch rectangle with the long side parallel to edge of counter, adding additional flour to counter and dough as needed to prevent sticking. Using a bench scraper or knife, trim dough into a 15-by-10-inch rectangle. Cut dough into six 5-inch squares (see note). Carefully transfer squares to parchment-lined baking sheet. Cover squares with second piece of parchment paper and refrigerate. Discard dough scraps or save for another use. (If using store-bought dough, see notes below.)

Repeat rolling and cutting with second rectangle of pie dough for 12 dough squares total. Cover baking sheet with plastic wrap and refrigerate until dough is well chilled, about 1 hour.

Meanwhile, in a 2-cup liquid measuring cup, whisk together apple juice, sugar, cornstarch, lemon juice, cinnamon, and salt until sugar and cornstarch are dissolved and mixture is smooth.

In a medium saucepan, melt butter over medium heat. Add apples and cook, stirring occasionally with a rubber spatula, until softened, 12 to 15 minutes.

Whisk juice mixture to recombine and pour over apples. Cook, stirring constantly, until thick and glossy, 1 to 2 minutes. Remove pan from heat. Transfer apple mixture to a shallow rimmed dish or pie plate and let cool to room temperature, about 45 minutes.

When ready to assemble hand pies, line second baking sheet with parchment. Arrange 6 chilled dough squares on lightly floured work surface. (Return remaining dough squares to refrigerator.) Brush edges of squares with prepared egg wash. Place 2 tablespoons cooled filling in center of each square, leaving clear ¼ -inch border around edges (do not overfill). Fold dough over filling to form rectangle. Using fingers, lightly press down to shape filling into an even layer, working from the center out, pressing out any air pockets. Press and crimp dough edges together to seal. (If dough squares become too soft, refrigerate until firm again.) Transfer assembled pies to parchment-lined baking sheet and refrigerate.

Remove remaining dough squares and repeat filling and sealing. Transfer to sheet with other pies. Loosely cover with plastic wrap and refrigerate for at least 30 minutes or up to 1 day.

When ready to fry, add oil to a large Dutch oven or pot until it measures about 2 inches deep and heat over medium heat to 350°F (175°C). Set wire rack in a rimmed baking sheet and line half of rack with a triple layer of paper towels. Using a clean, dry pastry brush, brush off any excess flour from pies. Holding one pie at a time in your hand, pinch the edges of each pie to ensure a tight seal.

When oil is hot, gently transfer 3 pies to oil (refrigerate remaining pies until needed). Cook until deep golden brown on both sides, using spider skimmer or slotted spoon to flip occasionally for even browning, 8 to 12 minutes. Adjust burner as necessary to maintain oil temperature around 350°F (175°C).

Transfer pies to paper towel–lined side of prepared rack. Let drain for about 30 seconds, then move pies to unlined side of rack. Return oil to 350°F (175°C) and repeat frying 3 pies at a time with remaining 9 pies (a total of 4 batches).

For Finishing the Pies: Optionally, in a medium bowl, combine sugar, cinnamon, and salt. Working with 1 to 2 pies at a time, toss in sugar mixture until coated and return to unlined side of rack (see notes). Let cool at least 5 minutes before serving. Serve warm or at room temperature