How to Make It
Combine butter, shortening, and sugar in the bowl of a stand mixer. Beat to combine, about 2 minutes. Add egg and vanilla and mix for another 2 minutes. In a separate bowl, whisk together flour, baking powder, baking soda, and salt. Add half of the dry ingredients to the wet ingredients with half of the milk; mix until just combined. Add remaining dry ingredients and milk, and beat until just incorporated. Scrape down sides of bowl ensuring that ingredients are evenly distributed. Wrap dough in plastic wrap and chill for 30 minutes.
Empty pie filling onto a cutting board, and use a large knife to chop the apple pieces until they are small chunks. Transfer filling to a bowl; set aside.
Preheat oven to 375°F. On a well floured surface, roll the dough out to 1/8-inch thickness. (Note: the dough will be very soft and sticky, so be sure to generously and regularly flour the surface under the dough if it begins to stick.) Using a 3-inch round cutter, cut out 14 circles. On a parchment paper-lined baking sheet, place 7 of the circles. Scoop 1 tablespoon of the apple pie filling into the center of each circle on the baking sheet, leaving a 1/4-inch ring around the circle with no apple filling. Lift the other circles from the work surface and place them directly on top of the circles with the apple filling. Gently press the sides together and use a fork to crimp around the edges. Reroll the remaining dough and repeat process of cutting, filling, stacking, and sealing cookie rounds. (Note: apple pie filling will be left over; either discard or reserve for another use.) Transfer baking sheet with cookies to refrigerator for 15 minutes.
Bake until just beginning to brown and dough has cooked through, 12 to 14 minutes. Allow to cool completely before icing.
Combine powdered sugar, cinnamon, and milk in a bowl. Stir, adding more milk if necessary until the icing is thick but easily spreadable. Spread a spoonful of the frosting on each of the cookies.