1. Combine flour, baking powder, cinnamon, and salt in a food processor, pulsing together until evenly combined. Add cubed butter and pulse in until the mixture is fine and crumbly. Add egg, canned pumpkin, and vanilla extract and pulse until a dough forms.
2. Shape the dough into a rough rectangle and wrapped in plastic wrap. Chill until firm, about 1 1/2 hours.
3. Preheat oven to 400 F and line two baking sheets with parchment paper. Liberally flour a surface for rolling out the dough. Coat the dough with a generous amount of flour as well. Cover the dough with a large sheet of parchment paper and roll out, adding more flour to the rolling surface and dough as needed in the process, to a rectangular shape 1/8" thick (the rectangle doesn't have to be perfect). Cut the rolled dough into shapes using cookie cutter of choice, transferring cut cookies to two prepared baking sheets. While rolling and cutting the dough, return to the fridge or freezer to chill if it gets too soft. On the other hand, if the dough cracks after chilling, let it soften a bit before continuing. I found it helpful to place the dough in the freezer for a few minutes after rolling, before cutting and moving to the baking sheet.
4. Once you've cut as many cookies as possible with the rolled-out dough, clump the leftover scraps in a ball, chill if needed, and reroll/cut shapes until all of the dough has been used up.
5. Once all cookies have been cut and moved to the prepared baking sheets, top with sprinkles (if desired). Bake in the preheated oven for 5-8 minutes, flipping and rotating sheets halfway through the baking time, until the cookies have puffed up and look and feel set on top. They shouldn't brown at all. Allow to cool before adding frosting.

To make the Brown Sugar Frosting

1. Microwave milk in a heat-safe container for about 20 seconds, until warm. Add brown sugar and stir in until completely melted.
2. Add the brown sugar mixture to the bowl of a stand mixer along with the softened butter. Using the beater attachment, beat on high speed until lighter in color and fluffier in texture. Add confectioner's sugar, mixing in slowly at first and increasing speed to high once it begins to get incorporated into mixture, beat until you have a soft, light frosting.
3. Use an icing spatula to spread a thin layer of frosting onto the tops of half of the cookies. Top each frosted half with another cookie.
4. Store cookies in an airtight container. Enjoy!