In a large bowl, whisk together the flour and salt. Cut in the cold butter until a crumb mixture forms.* Drizzle in 5 tablespoons of the water and mix until the dough comes together. Add an additional tablespoon of water, if needed. Gather the dough into a ball and wrap it with plastic wrap. Chill the dough for 1 hour before rolling.
In a large bowl, toss the apples and cranberries with the lemon juice. In a small bowl, mix the sugar, flour, cinnamon, and salt. Add the sugar mixture to the fruit mixture and toss to combine.
To assemble the hand pies, line a large baking sheet with parchment paper or Silpat mats, or spray with cooking spray.
Remove the dough from the refrigerator. Lightly dust a work surface with flour. Roll out the dough to ¼-inch thickness. Cut the dough using a 5-inch round cutter, gently rerolling the dough scraps as needed.*
Scoop 2-3 tablespoons of the filling mixture onto one side of each round. Rub a little bit of water around the edge of each round, then fold the rounds over so the edges meet and use a fork to seal the edges.
Transfer the hand pies to the baking sheet, leaving 1 ½ inches between each pie. Beat the egg white and water together in a bowl to make an egg wash. Use a knife to cut 3 slits into the top of each pie, brush with the egg wash, and sprinkle with sugar. Chill for 30 minutes.
Preheat the oven to 375 degrees.
Bake the pies until they are golden brown throughout, for 30-35 minutes. Let the pies cool for 10 minutes on the baking sheet and then transfer them to a wire rack to cool completely. The pies may be made up to 1 day ahead.