Sift flour and salt together. Cut in the shortening with a pastry blender until the mixture resembles coarse crumbs. Add cold water 1 tablespoon at a time and mix with fork. When the flour mixture is moistened, gather it into a ball, wrap it in plastic, and refrigerate for 30 minutes.
Peel and dice the apples and place them in a saucepan. Combine the sugar and cinnamon; pour over the apples and toss to coat. Cook, covered, in a saucepan on low heat. Cook until soft, then mash with fork to form a thick applesauce. Allow to cool.
Transfer the dough to a lightly floured work surface. Roll the dough out to 1/8 inch thick and cut rounds with a large cookie cutter (4 inches in diameter).
In each round, place 1 heaping tablespoon fruit. Moisten edges with cold water, fold in half, and press edge with a fork to seal. Repeat with the remaining pastry and filling.
Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C).
Fry the pies, a few at a time, 2 to 3 minutes on each side; cook until the crust is golden brown. Drain on paper towels.