Special equipment: a 2 1/2-inch round biscuit cutter about 20 wooden ice-pop or lollipop sticks a pastry brush
For the pie crust: Pulse the flour, shortening, sugar, vinegar and salt in a food processor until it looks like fine meal. Add the butter and pulse until it is in pea-size pieces. Sprinkle in 1/4 cup ice water and pulse until the dough begins to come together. Pinch the dough with your fingers; if it doesn't hold together, add up to 4 more tablespoons ice water 1 tablespoon at a time, pulsing and checking the dough after each addition. Divide the dough between 2 sheets of plastic wrap and pat each into a disk. Wrap tightly and refrigerate until firm, at least 1 hour or preferably overnight.
For the apple pie filling: Peel and core the apples; slice 1/4 inch thick. Transfer to a bowl and toss with the sugar and lemon juice.
Melt the butter in a medium skillet over medium-high heat. Add the apple mixture and cook, stirring occasionally and breaking up with a wooden spoon, until very soft, 12 to 15 minutes. Add the flour, cinnamon and salt and cook, stirring, until the juices thicken, about 2 minutes. Let cool completely.
Position 2 oven racks in the top and bottom third of the oven and preheat to 400 degrees F. Line 2 large baking sheets with parchment paper and set aside. Put the sugar in a shallow bowl and set aside.
To assemble: Roll out 1 disk of dough on a lightly floured surface to 1/8 inch thick. Cut into about 20 rounds with a 2 1/2-inch biscuit cutter, keeping the dough and rounds cool and rerolling scraps once if neccessary. Repeat with the second disk, for a total of about 40 dough rounds.
Divide half of the dough rounds (about 20) between the baking sheets, leaving 1 inch of space around each. Dip the top 1 inch of a wooden ice-pop stick in the beaten egg and lay it on one of the rounds, positioning the tip in the center of the round. Repeat with the remaining sticks and dough rounds on the baking sheets. Dollop a generous teaspoonful of the cold apple filling in the center of each round, covering the tip of each stick. Brush the edges of the rounds with the beaten egg using a pastry brush. Put the remaining dough rounds over the rounds with the filling. Press the edges together gently to seal. Use a fork to crimp the rounds closed completely, making pops. Brush the tops with the remaining beaten egg and dust each pop with a generous pinch of sugar. Using a paring knife, make 5 small slits around the centers of each pop (they'll look like apple seeds).
Bake, reversing the baking sheets top to bottom halfway through, until the dough is golden brown, 18 to 22 minutes. Let the pops cool on the baking sheets until cool enough to touch, about 5 minutes. Transfer to a wire rack to cool. Serve the pops warm or at room temperature.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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