For the lolli-pie dough: Mix the flour, sugar and salt in a food processor fitted with a steel blade. Scatter the butter pieces over the flour mixture, tossing to coat the butter with a little of the flour. Cut the butter into the flour with five 1-second pulses. Add the shortening and continue cutting in until the flour is pale yellow and resembles coarse cornmeal, with butter bits no larger than small peas, about 4 more 1-second pulses. Turn the mixture into a medium bowl.
Sprinkle 3 tablespoons ice water over the mixture. With the blade of a rubber spatula, use a folding motion to mix. Press down on the dough with the broad side of the spatula until the dough sticks together, adding up to 1 tablespoon more ice water if it will not come together. Shape into a ball with hands, then flatten into a 4-inch-wide disc. Dust lightly with the flour, wrap in plastic and refrigerate at least 30 minutes before rolling.
For the pumpkin chai lolli-pie filling: Preheat the oven to 350 degrees F. Soak the popsicle sticks in water for at least 30 minutes. Dry well before using.
Whisk together the cream cheese, eggs and pumpkin using an electric mixer. In a separate bowl, mix together the remaining dry ingredients. Add the dry ingredients to the wet and thoroughly incorporate. Bake the mix in a small baking dish until set (pumpkin mixture will be somewhat firm and should not jiggle when shook), about 15 minutes.
For assembling the pumpkin chai lolli-pies: Remove the dough from refrigerator. If stiff and very cold, let stand until the dough is cool but malleable. With a floured rolling pin, roll the dough disc on a lightly floured surface until 1/4-inch thick. Using a 3-inch biscuit cutter, cut 40 circles from the dough. Keep 5 discs on your work surface and refrigerate the remaining disks on a baking sheet.
Set a popsicle stick on each disc such that one end of it is resting in the center. Spoon about 2 teaspoons of the pie filling into the center of each disc. Remove 5 more discs from the refrigerator and center one on top of each disc that has pumpkin filling. Using the tines of a fork, bind the 2 discs together, impressing around the edges of each pie pop. Follow suit, making 15 more lolli-pies, using the remaining discs of dough.
Brush both sides of each lolli-pie with a wash of the egg yolk whisked with 1 teaspoon water. Sprinkle heavily with sugar. Set on a parchment paper-lined baking sheet and bake about 7 minutes. Flip the lolli-pies and bake until golden brown, 7 to 8 minutes longer. Remove from the oven. Allow to cool. Serve.
This recipe was provided by a professional chef or restaurant and may have been scaled down from a bulk recipe. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.