Heat oven to 275 degrees. Place two racks in the oven. Beat the butter on medium-low speed until smooth, 2 to 3 minutes.
Beat in sugar and vanilla on medium. Sift together flour, salt, and green tea. Gradually beat into butter mixture on low. Transfer dough onto a piece of plastic. Pat into 8-inch disk. Wrap with plastic; chill 1 hour. May be made a day in advance.
On a lightly floured surface, roll out dough to 1/4 inch thick; cut into 1-to- 2 1/2-inch rounds. Reroll until all dough is used. Press each round with tines of fork or bowl of spoon to create decoration. Transfer to two unlined baking sheets. Chill 15 minutes before baking. Bake 25 to 30 minutes until firm but not brown; let cool. Cookies can be made 2 to 3 days ahead, and stored in an airtight container.
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