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Special equipment: a 2 1/2-inch round biscuit cutter, about 20 wooden ice-pop or lollipop sticks and a pastry brush
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For the pie crust: Pulse the flour, shortening, sugar, vinegar and salt in a food processor until it looks like fine meal.
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Add the butter and pulse until it is in pea-size pieces.
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Sprinkle in 1/4 cup ice water and pulse until the dough begins to come together.
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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.
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Divide the dough between 2 sheets of plastic wrap and pat each into a disk.
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Wrap tightly and refrigerate until firm, at least 1 hour or preferably overnight.
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For the apple pie filling: Peel and core the apples; slice 1/4 inch thick.
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Transfer to a bowl and toss with the sugar and lemon juice.
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Melt the butter in a medium skillet over medium-high heat.
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Add the apple mixture and cook, stirring occasionally and breaking up with a wooden spoon, until very soft, 12 to 15 minutes.
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Add the flour, cinnamon and salt and cook, stirring, until the juices thicken, about 2 minutes.
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Let cool completely.
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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.
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Put the sugar in a shallow bowl and set aside.
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To assemble: Roll out 1 disk of dough on a lightly floured surface to 1/8 inch thick.
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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.
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Repeat with the second disk, for a total of about 40 dough rounds.
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Divide half of the dough rounds (about 20) between the baking sheets, leaving 1 inch of space around each.
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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.
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Repeat with the remaining sticks and dough rounds on the baking sheets.
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Dollop a generous teaspoonful of the cold apple filling in the center of each round, covering the tip of each stick.
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Brush the edges of the rounds with the beaten egg using a pastry brush.
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Put the remaining dough rounds over the rounds with the filling.
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Press the edges together gently to seal.
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Use a fork to crimp the rounds closed completely, making pops.
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Brush the tops with the remaining beaten egg and dust each pop with a generous pinch of sugar.
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Using a paring knife, make 5 small slits around the centers of each pop (they'll look like apple seeds).
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Bake, reversing the baking sheets top to bottom halfway through, until the dough is golden brown, 18 to 22 minutes.
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Let the pops cool on the baking sheets until cool enough to touch, about 5 minutes.
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Transfer to a wire rack to cool.
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Serve the pops warm or at room temperature.
