Sift flour, tea powder, and salt into a small bowl; set aside. Place butter in the bowl of an electric mixer fitted with the paddle attachment. Cream on medium speed until fluffy, 3 to 5 minutes. Add sugar; continue to beat until very light in color and fluffy, about 2 minutes more. Add flour mixture; combine on low, scraping sides of bowl with a spatula if necessary, until flour is just incorporated and dough sticks together when squeezed with fingers.
Place a piece of parchment on a clean surface; dust with flour. Roll dough to 1/4-inch thickness; chill in refrigerator or freezer until firm, about 30 minutes.
Preheat oven to 325 degrees. Line two baking sheets with parchment. Cut chilled dough with 2-inch leaf cutters. Using a wide spatula, transfer to baking sheets. Chill until firm. Gather scraps together, re-roll, chill, and cut shapes. Bake until firm and barely starting to color, 15 to 20 minutes, rotating halfway through. Cool completely on wire rack; store in an airtight container for up to 3 to 4 weeks.