In a medium mixing bowl, beat the room-temperature butter with the granulated sugar on medium-high speed using a hand mixer until light and creamy, about 3 minutes. Add the egg yolk and vanilla, then beat until well combined. If using a stand mixer, scrape down the sides of the bowl a few times.

Sift the flour, cornstarch, baking powder, pumpkin pie spice, and salt into the butter mixture. Mix on low speed until a crumbly dough starts to come together.

Add the milk one teaspoon at a time, mixing just until a dough ball forms and pulls away from the sides of the bowl. The dough should feel dry yet slightly sticky. Use only as much milk as needed, since humidity affects the texture.

Gather the dough into a ball and place it on a sheet of parchment paper set on a baking sheet. Roll the dough to just under 1/4-inch thickness. Cover with plastic wrap and refrigerate for at least 2 hours, or up to 2 days.

Preheat the oven to 350°F and position the rack in the middle.

Remove the baking sheet from the fridge and cut out shapes using cookie cutters. Save the dough scraps for re-rolling. Bake the cookies for 12 to 13 minutes, or until the bottom edges just start to turn light golden. If the dough sticks to the rolling pin, dust the top lightly with flour.

While the first batch bakes, re-roll the dough scraps on a second piece of parchment paper placed on a baking sheet, then cut out more cookies. Loosely cover and refrigerate until ready to bake, then bake for 12 to 13 minutes.

Cool the cookies on the baking sheets set over wire racks for 10 minutes, then transfer them to the racks to cool completely.

Once the cookies are completely cool, whisk the powdered sugar and maple syrup together in a small bowl until the sugar is moistened. Add milk or water a teaspoon at a time, stirring until the icing reaches the consistency you prefer for spreading or piping.