Cut open the tea bags and pour the leaves into a small food processor with the granulated sugar. Pulse until the tea is very finely ground to create 'tea sugar'.

In a medium bowl, whisk together the cake flour, cornstarch, and salt.

In a mixer, beat the softened butter with 1/2 cup of the tea sugar until light and fluffy. Add the dry ingredients and vanilla extract. Mix on low speed just until a dough forms.

Divide the dough in half, press each half into a disk, wrap well, and refrigerate for 20 minutes.

Preheat the oven to 325°F and line baking sheets with parchment paper.

Roll one disk to about 1/4 inch thick on a lightly floured surface and cut into 2-inch squares or rounds. Re-roll scraps once. Place cookies at least 2 inches apart, prick each with a fork, and chill for 10 minutes.

Sprinkle remaining tea sugar over the cookies, then bake in batches for 12 to 15 minutes until the bottoms are pale golden.

Cool cookies completely on a wire rack before storing.