Whisk the flour, salt, and 1 tbsp granulated sugar together; cut in the shortening until crumbly, then add cold water a little at a time until the dough comes together. Wrap and refrigerate for 30 minutes.

Cook the diced apples with both sugars, cinnamon, and nutmeg over medium heat for 6 to 8 minutes; stir in the cornstarch slurry and cook 1 more minute until thickened, then stir in the butter. Cool for 15 minutes.

Roll the dough to 1/8-inch thickness and cut into 5 to 6-inch circles; fill each with 2 tbsp of the cooled apple filling, fold into half-moons, and press the edges firmly with a fork.

Heat the vegetable oil to 350°F (175°C) and fry the pies in batches of 2 to 3, about 3 to 4 minutes per side, until deep golden brown. Drain on a wire rack for 2 minutes.

Arrange on a plate, dust with powdered sugar, and serve warm with one pie cut open to show the filling.