In the bowl of a food processor combine the flour, salt and sugar. Add the cubes of butter and pulse until the mixture is crumbly. Slowly pour ¼ cup of the ice water over the flour mixture and pulse until the mixture forms a dough. Add additional water, 1 teaspoon at a time if needed.
Turn the dough out onto a clean, lightly floured surface and gather with your hands. Form the dough into a disk and wrap in plastic wrap and refrigerate for 30 minutes.
For the filling:
Heat a large sauté pan over medium heat, add the apples, brown sugar, butter, lemon juice, cinnamon and cornstarch, salt and 2 tablespoons (30g) of water. Cook, stirring occasionally, until the apples are tender, about 8 to 10 minutes.
Transfer the mixture to a bowl and let it cool completely.
For assembly:
Heat oven to 350ºF (180ºC). Divide the dough in half and roll each half out into a rectangle about ¼” thick.
Cut each rectangle into 8 pieces so each is 4” x 3”. You may have to re-roll the scraps to make enough. Lay the pieces on a parchment paper-lined baking sheet. Repeat with the other half of the dough.
Use a slotted spoon to mound 2 tablespoons of the apple pie filling into the center of the dough being sure to drain as much of the liquid as possible.
Top with one of the remaining rectangles of dough and use your fingers to gently press the edges together. Use the back of a fork to seal the edges together tightly. Repeat with the remaining pies.
In a small bowl whisk together the egg and milk until combined. Use a pastry brush to brush each of the hand pies with the egg wash mixture.
In another small bowl stir together the sugar and cinnamon and top each pie with a sprinkle of the mixture. Use a knife to cut three small slits in the top of each.
Bake for 15 minutes then take out and sprinkle with the cheese over the top. Bake until golden brown and the cheese is melted, about 10 minutes.
Notes
The salty sharpness of the cheese balances the sweetness of the filling beautifully. If cheese on pie feels like too much, try a milder cheddar or skip the topping!