In a large bowl, whisk together the gluten free flour, cornstarch, baking powder, and kosher salt. Set aside.
In a small bowl, mix 2 rounded tablespoons of cornstarch with 3 tablespoons of water using a spoon. Stir until thin and watery. Set aside.
In a large mixing bowl, beat the room temperature butter and 1¼ cups of granulated sugar with an electric mixer for 2-3 minutes until soft and creamy.
With the mixer on low, slowly add in the cornstarch water, milk, vanilla extract, and 3.5 tablespoons of the chilled strawberry syrup. Mix until combined. The mixture will look pale pink and slightly chunky.
Slowly beat in the flour mixture a little at a time until just combined. If using red food coloring, add it now and mix until evenly blended throughout the dough.
Cover the bowl with plastic wrap and refrigerate for at least 2.5 hours. Do not skip this step or the cookies will spread flat.
When the dough is almost done chilling, preheat the oven to 325°F (163°C) and line a baking sheet with parchment paper. Do not grease.
Using a large cookie scoop (2 oz), scoop 12 balls of dough. If the dough sticks to the scoop, rinse the scoop halfway through.
Roll each dough ball completely in the remaining 2 tablespoons of granulated sugar.
Place dough balls 3-4 inches apart on the prepared baking sheet. Bake at 325°F for 15 minutes. Keep remaining dough balls in the fridge while each batch bakes.
Immediately after removing from the oven, use a spatula to gently reshape the cookies into circles. Let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.