Preheat your oven to 350°F and line two large rimmed baking sheets with parchment paper or a silicone baking mat (don’t grease).
In a medium bowl, whisk the flour, cornstarch, baking powder, and salt and set aside.
In a large bowl, use a hand-held electric mixer to cream the softened, unsalted butter with the sugar until light and fluffy. Add the egg, egg yolk, and vanilla extract and mix until combined, scraping down the sides of the bowl with a rubber spatula as needed. Add the dry ingredients and mix until combined. The dough will be very thick.
On a lightly floured and clean surface, roll the dough out to ¼" thick. Lightly dust the dough and your rolling pin with flour as needed. Cut out the cookie shapes and arrange them on the prepared baking pan. Gather up the dough scraps, re-roll, and repeat until all of the dough is used. Sprinkle the dough and your rolling pin with more flour as needed to prevent the dough from sticking. Do your best to get as many cookies cut out as you can each time you roll the dough out so you don’t end up adding too much flour.
Bake in the preheated oven for 12-14 minutes, or until the edges are just starting to turn light golden brown. Remove from the oven and let cool slightly on the pan, and then transfer to a plate or a cooling rack to cool completely.
Make the cookie icing: In a medium bowl, whisk the powdered sugar, light corn syrup, heavy cream, vanilla extract, and salt. If the icing is too thick you can add a little more milk or cream, or if it's too thin you can add a little more powdered sugar. Divide the icing into separate bowls and add more or less red food coloring to each to make different shades of red or pink. Gel food coloring is best because you don't need as much of it. Liquid food coloring can make the icing runny, so you might need to add a little more powdered sugar to thicken it up.
Frost and decorate the cooled cookies as desired. If adding sprinkles or other decorations be sure to do so before the icing sets.
Store cookies (frosted or unfrosted) in an airtight container at room temperature for 1 week, or in the fridge for 10 days.