Preheat oven to 350 degrees. Butter a 13x9-inch baking dish; set aside.
Peel, core, and cut each of the apples into 8 pieces, giving you 16 total apple slices. Place apple slices into a bowl of cold water with a few tablespoons of lemon juice. This will prevent the apples from browning while you work.
In a large bowl, whisk together the flour, salt, and 2 tablespoons of sugar. Using a pastry blender or two knives, cut in the shortening and the butter until the mixture resembles coarse meal. Pour in the cold buttermilk and stir together with a fork until the dough is shaggy but just starting to come together.
Turn the dough out onto a floured surface and knead 2 or 3 times, just to bring the dough together. Form into a disk and cut it into 16 equal pieces. Pat each piece into a circle approximately 4-inches in diameter. Place an apple slice in the center of the dough (make sure to shake off any excess water), and fold the dough around the apple, pinching to seal the edges. Arrange dumplings in the prepared baking dish.
Melt the stick of butter. Whisk in the ¾ cup of sugar and vanilla – the mixture will be thick and grainy. Pour evenly over the dumplings, and then pour the ginger beer around the edges of the pan. There will seem to be a lot of liquid around your dumplings – this is ok! It will reduce into a lovely, thick syrup in the oven.
Whisk together the cinnamon and sugar. Sprinkle evenly over the top of the dumplings and liquid. (This will ultimately form a nice crust on the top of the dumplings.)
Bake dumplings for 35-45 minutes, or until nicely golden on top and the syrup is bubbly. Allow to cool for several minutes before serving warm.
Will keep for 3 days, covered in the refrigerator.
All butter may be used in the pastry dough in lieu of the butter-shortening combination. Ginger beer can usually be found near the root beer at your grocery store.