Preheat the oven to 350°F/180°F (160°C fan oven), and line 2 cookie sheets with parchment paper or silpat silicone mats.
In a mixing bowl, combine flour with baking soda, baking powder, and salt. Set aside.
In a separate medium mixing bowl, using a hand mixer cream butter with sugar until smooth and creamy (3 minutes).
Add egg and vanilla, and beat again for 1 minute. Scrape down bowl as needed.
Add the dry ingredients to the wet ingredients, and mix until just combined.
Form cookie dough balls with a medium cookie scoop (1.5 tablespoon size), and place on the prepared cookie sheets.
Spray the bottom of a clean glass with baking spray, and press each cookie until it's about 1⁄2-inch thick.
Bake in the preheated oven for 7-8 minutes or until the centers of the cookies are set. Allow to cool on the cookie sheets for 10 minutes, then transfer to a wire rack to cool completely.
To Make the Icing:
In a bowl, whisk together the powdered sugar with milk, corn syrup if using, and vanilla extract. Add more milk or more powdered sugar if needed to reach the desired consistency.
Split the icing into two small bowls, and add food coloring to each.
Dip the top of each cookie into the prepared icing, and allow the excess to drip off, then place on a wire rack and sprinkle with crushed red mints and allow to set/dry.
Notes
Don't Flatten too Much: These cookies spread a little, so make sure not to flatten them too much before baking.
Overbaking: Remember that all ovens are different, so you have to watch your cookies to make sure not to overbake them. Overbaked cookies are dry and crumbly, and you want to avoid that.
Corn syrup helps the icing harden. If you don't have it, you can leave it out.
Storing: When the icing has set, store the cookies in a single layer in an airtight container for up to 1 week.
Freezing: These cookies freeze well in an airtight container for up to 3 months.