Preheat the oven to 350 degrees F and line a 9-by-13-inch baking dish with foil, leaving a 2-inch overhang on all sides. Pulse the tea leaves briefly in a spice grinder until ground but not powdery. Transfer to a medium bowl and whisk in the flour and salt.
Beat the butter in a large bowl with a mixer on medium-high speed until fluffy, about 2 minutes. Reduce the mixer speed to low; add 2 1/2 cups confectioners¿ sugar and the vanilla and beat until incorporated, then increase the speed to medium high and beat until smooth, about 1 more minute. Reduce the mixer speed to low and beat in the flour mixture, then increase the speed to medium and beat until just combined.
Scrape the dough into the prepared pan. Lay a piece of plastic wrap on top and press the dough evenly into the pan; remove the plastic wrap. Bake until the shortbread is firm and the edges are just light golden, 40 to 45 minutes. Transfer to a rack and let cool 20 minutes in the pan. Gently lift the shortbread out of the pan using the foil overhang. Slice into 1 1/2-inch squares. Transfer to racks to cool completely.
Make the glaze: Whisk the remaining 1 cup confectioners¿ sugar and the milk in a bowl until smooth; drizzle over the shortbread. Sprinkle with coarse sugar and let set, 20 to 30 minutes.