In a food processor, pulse together the flour, tea, and salt, until the tea is just spotted throughout the flour. Add the confectioners' sugar, vanilla, and butter. Pulse together just until a dough is formed. Place dough on a sheet of plastic wrap, and roll into a log, about 2 1/2-inches in diameter. Tightly twist each end of wrap, and chill in refrigerator for 30 minutes.
Preheat oven to 375 degrees F.
Slice the log into 1/3-inch thick disks. Place on parchment or silpat lined baking sheets, 2 inches apart (2 probably needed depending on size of sheets). Bake until the edges are just brown, about 12 minutes. Let cool on sheets for 5 minutes, then transfer to wire racks and cool to room temperature.