Preheat your oven to 350°F (177°C) and line two baking sheets with parchment paper.

In a large mixing bowl, whisk together all-purpose flour, cornstarch, baking powder, baking soda, salt, and culinary-grade matcha powder until well combined.

In a separate bowl, beat the softened unsalted butter and granulated sugar together for 2-3 minutes until light and fluffy.

Add the egg, sour cream, and vanilla extract to the butter mixture and mix until smooth.

Gradually add the dry ingredient mixture to the wet ingredients and mix on low speed until a soft dough forms.

Scoop out 2-tablespoon portions of dough, roll into balls, flatten slightly, and chill in the refrigerator for at least 20 minutes.

Bake the chilled dough in the oven for 9-11 minutes until edges are set and centers look slightly underdone.

Remove from the oven and cool on baking sheets for about 5 minutes, then transfer to a wire rack.

Heat the whole milk or heavy cream and steep the culinary lavender buds for about 10 minutes. Strain to remove buds and beat with softened butter and powdered sugar until smooth.

Gradually add the lavender-infused milk until reaching desired consistency. Frost cookies once cool.