Preheat the oven to 350°F (175°C). Line a large baking sheet with parchment paper.
Beat the softened butter and granulated sugar in a large bowl until light and fluffy. Mix in the egg and vanilla extract until well combined.
Stir in the flour, cornstarch, baking powder, and salt. Mix until a soft dough forms. If the dough feels too sticky, chill it in the refrigerator for 20 minutes.
Roll the dough out on a lightly floured surface to 1/4-inch thickness. Cut out 12 circles using a 2-inch or 3-inch round cookie cutter. Reroll scraps as needed.
Place the cutouts on the prepared baking sheet, leaving 2 inches between each cookie. Bake for 12 to 15 minutes, or until the edges are light golden brown. Let the cookies cool completely on a wire rack.
Whisk the powdered sugar and 2 tablespoons of water in a medium bowl to make the icing. Add more water a few drops at a time until the mixture resembles thick glue.
Separate about 4 tablespoons of the icing into four small bowls. Dye one black, one orange, one brown, and one red using the gel food coloring. Transfer these colored icings to separate piping bags with small tips.
Pipe black eyes and mouths and orange noses onto the sides of the 12 marshmallows. Spoon a small puddle of plain white icing onto each cooled sugar cookie.
Press a decorated marshmallow head onto the back edge of each white icing puddle. Place three mini chocolate chips in a row on the icing for buttons. Let the icing set for 15 minutes, then pipe brown arms and a red scarf onto the snowmen.