For the pie crust, mix together the flour, salt and sugar in the bowl of a stand mixer with the paddle attachment. With the paddle running, slowly add the butter cubes one by one, until the mixture resembles a coarse meal. Add the tablespoons of water one at a time, or until the mixture comes together. Let chill in the refrigerator for at least 30 minutes.
To make the filling, heat the butter in a large pan (or even a large pot) over medium heat. Add the apples, sugars, cinnamon, and nutmeg. Let the mixture cook for about 5 minutes, or until the apples soften. Remove the apples with a slotted spoon to a separate bowl. Keep the leftover juices in the pan. *If you're making the caramel, at this point let the apple juices simmer in the pan on medium low heat for about 20 minutes, stirring occasionally.
Take half of the dough and roll out onto a lightly floured surface, until about 1/4" thick. Use a circular object to cut rounds out (I used a large drinking glass). Whisk an egg in a separate bowl and have a pastry brush ready.
Place a heaping tablespoon (or less, dependent on how big your dough circles are) of the filling in the center of a circle. Dip your pastry brush in the egg and brush it around the edges. This helps the top layer of dough stick. Take a second dough circle and place on top of the one with filling, stretching a bit to reach over the apples. Use your fingers and a fork to crimp and seal the edges.
Repeat this process until you run out of dough and/or filling. Only re-roll the dough once. Place the pies on a cookie sheet, and brush the tops with the egg. With a small, sharp knife, cut two small slits in the center for steam to escape. Sprinkle with white sugar. Bake at 350 degrees Fahrenheit for 20 minutes, or until the edges are golden.
For the caramel (optional), transfer the apple juice and sugar mixture that has been simmering to a sauce pan and add the butter. Bring to a boil for 2 minutes over high heat. Add the cream and boil for another minute, or until it reaches your desired consistency. Serve in a small bowl for dipping or drizzling over the top of the hand pies.