PREHEAT oven to 350F. Position racks in top and bottom thirds of oven. Line 2 baking sheets with parchment.
FINELY grind Earl Grey tea in a coffee grinder. Stir flour with ground tea leaves and salt in a medium bowl.
BEAT butter with sugar and vanilla in a large bowl, using a wooden spoon, until smooth, 1 min. Gradually stir in half of flour mixture, until just combined. Add remaining flour mixture, kneading with your hands, until dough forms a ball.
ROLL out on a floured counter until 1/4 in. thick. Cut out cookies with various shapes of cookie cutters. Arrange on prepared sheets, 2 in. apart. Cookies will spread as they bake.
BAKE in top and bottom thirds of oven, switching sheets halfway through, until edges are just golden, 12 to 15 min. Leave on sheet for 5 min, then transfer cookies to a rack to cool completely. Cookies will keep well in an airtight container up to 1 week, or freeze up to 1 month.