Line two cookie sheets with parchment paper or silicone baking mats. Set aside until needed.
First, make the cookies. Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment, and beat on medium speed until light and fluffy, scraping the sides and bottom of the bowl as necessary. Add in the cornstarch, vanilla extract, and almond extract, and beat again until smooth.
In a medium bowl, whisk together the flour, baking powder, and salt, then add half the dry ingredients to the butter mixture. Scrape the bowl again, then mix together on low speed until the flour mixture is well-incorporated. Add the rest of the flour mixture and the milk, scraping the bowl again as needed, and mix again until a soft, pliable dough forms.
Place one sheet of parchment paper (from your cookie sheets) on a clean work surface and dust it lightly with flour. Put the cookie dough in the center, press it down into a flattened disc shape, then dust with flour. Use a rolling pin (a straight-sided, sturdy drinking glass will work, too) to roll the dough out to a thickness between ¼ inch and ⅜ inch: the thicker the dough, the softer the cookies! Once rolled out, transfer the piece of parchment back to the cookie sheet and place the entire thing in the refrigerator to chill for an hour. Alternatively, freeze the rolled-out dough for 15 to 20 minutes.
When ready to bake the cookies, preheat the oven to 350°F. Remove the chilled dough from the refrigerator or freezer and cut out shapes using your favorite cookie cutters. Transfer the cookies to the other parchment-lined cookie sheet, leaving at least an inch between cookies. Re-roll and cut the remaining cookie dough until all the dough has been used.
Transfer the cookies to the preheated oven and bake for 12 to 14 minutes, until lightly golden brown. Remove from the oven, allow to cool on the cookie sheets for 5 minutes, then transfer to cooling racks to cool completely.
While the cookies are cooling, make the frosting. Place the vegan butter in the bowl of a standing mixer fitted with the whisk attachment, and beat on medium speed until fluffy. Add in half the powdered sugar and mix on low speed until the sugar is incorporated. Scrape down the sides of the bowl with a spatula, then add the rest of the powdered sugar, the vanilla and almond extracts, and 1 tablespoon non-dairy milk and beat again, starting on low speed and working up to medium-high speed until the buttercream is light and fluffy. Add an additional teaspoon or two of milk if the buttercream is too thick to spread and mix again. If using food coloring, scrape a bit onto a toothpick, swirl into the frosting, and beat again. Add additional food coloring until the desired shade has been reached.
Spread each cookie with a generous layer of frosting, garnishing with sprinkles if you like. Serve immediately, or store in a covered container in the refrigerator for up to 5 days.
