Preheat your oven to 350°F (177°C) and prepare baking sheets by lining them with parchment paper.

In a medium bowl, whisk together all-purpose flour, cornstarch, baking powder, baking soda, salt, and matcha powder.

In a large bowl, beat softened butter and granulated sugar together until creamy, about 2-3 minutes.

Mix in the egg, sour cream, and vanilla extract to the butter-sugar blend until smooth.

Gradually fold the dry ingredient mixture into the wet mixture until a thick dough forms.

Scoop and roll the dough into 2-tablespoon-sized balls, slightly flattening the tops. Chill in the refrigerator for at least 20 minutes.

Bake for 9 to 11 minutes until edges are set and centers appear slightly underdone.

Cool the cookies on the baking sheets for a few minutes, then transfer to a wire rack to cool completely.

To make the frosting, heat milk and steep lavender buds in it. Strain and mix with softened butter and powdered sugar until smooth.

Frost each cookie generously with lavender-infused frosting and add food coloring if desired.