For the dough: Sift the flour, sugar, and salt into a large bowl. Add the butter and shortening. Rub the fats between your fingertips and thumbs, or use a pastry blender, until the mixture resembles coarse crumbs. One tablespoon at a time, work in enough ice water just until the dough comes together, being careful not to over-mix.
Shape into a smooth ball of dough, flatten into a disk, and wrap in plastic wrap. Refrigerate for at least 30 minutes and up to 2 days.
When ready to use, sprinkle a little flour on a smooth work surface, and using a rolling pin or dowel, roll the pie dough into a large circle about 1/8-inch thick. Use a cookie cutter or small knife to cut 6 (6-inch) rounds into the dough. Place the pie dough rounds on a parchment lined baking sheet and place in the refrigerator to cool.
For the filling: In a large saute pan set over a medium-high heat, add the butter and melt. Place the apples in the saute pan and cook for 1 minute. Add the sugar, cinnamon, and rum to the pan and cook for 2 minutes more. While the apples are cooking, make a slurry out of the cornstarch and lemon juice and add it to the apples. Continue to cook for 1 minute and remove from the heat. Place the apples on a parchment lined sheet pan to cool.
When the apples are sufficiently cool enough to handle, remove the pie dough rounds from the refrigerator. Divide the apples evenly between the rounds. Be sure to place the apples on only half of the round to allow space to fold the other half over the first. Once you fold the dough over, pinch the edges together and then crimp the edges with the tines of a fork. Make a slit in the top of the turnover to act as a vent for the steam. Whisk the egg yolk and milk together and brush the turnovers with the egg wash. Place the turnovers in the refrigerator to cool. Preheat the oven to 400 degrees F.
When the oven is hot, put the turnovers in the oven and bake for 25 minutes, or until golden brown. Remove from the oven and serve either hot, or when cooled. Serve with Banana Chocolate Ice Cream.