In a large bowl, whisk together the flour, cornstarch, baking powder, baking soda and salt.
In a separate large bowl, beat together the butter, cream cheese and sugar until creamy.
Mix in the vanilla extract, peppermint extract and egg. Turn off the mixer and scrape down the sides of the bowl.
Add the flour mixture into the butter mixture. Start mixing with the electric mixer on a low speed, then gradually increase to medium speed until all the dry ingredients are combined.
Turn off the mixer. Squeeze the dough between your fingers. If it sticks to your fingers, mix in 1-3 more tablespoons of flour until it no longer sticks.
Divide the dough in half. Form each half into a ball and then flatten into a round disc about ½ inch (1 cm) thick and wrap in plastic. Chill in the fridge for at least 4 hours or up to 48 hours.
When ready to bake, preheat the oven to 350F or 325F on a convection oven (180C or 170C) and line cookie sheets with baking paper or baking mats (this helps prevent the bottoms from burning or over browning).
Take the dough out of the fridge. If it's been in there overnight, it may need to sit on the counter for 5-10 minutes first.
Lightly flour a clean work surface and dust a rolling pin with flour. Unwrap 1 disc from the plastic and gently roll out until it's about ¼ - ⅓ inch (about ½ cm) thick. Cut into shapes using a round cookie cutter and place cookies 1.5 inches apart on the lined cookie sheets. Repeat the process until all the dough is used.
Bake 1 cookie sheet at a time in the preheated oven for about 7-10 minutes, or until the tops no longer look doughy. The cookies should not be golden brown. Cool completely.