In large bowl, whisk together flour, baking powder, and salt. Add contents of tea bags; whisk until well combined. Set aside.
In a separate bowl and using an electric mixer, cream butter and sugar until light and fluffy. Beat in egg and vanilla. With mixer on low, gradually add flour mixture; beat until combined. Divide dough in half; flatten into disks. Wrap each in plastic; freeze until firm, at least 20 minutes.
Preheat oven to 325°F. Line baking sheets with parchment paper.
Remove one dough disk; let stand 5 to 10 minutes. Roll out ⅛ inch thick between two sheets of floured parchment, dusting dough with flour as needed. Cut shapes with cookie cutters. Using a spatula, transfer to prepared baking sheets. (If dough gets soft, chill 10 minutes.) Reroll scraps; cut shapes. Repeat with remaining dough.
Bake, rotating halfway through, until edges are golden, 10 to 18 minutes (depending on size). Cool completely on wire racks.