Cream butter and confectioners sugar in a large bowl or stand mixer fitted with a paddle attachment.
Add in flour, salt, and loose tea leaves. continue to mix until a soft dough forms, the mix will be dry and crumbly at first but will come together.
Roll out dough on a lightly floured surface until it's ¼ inch thick.
Cut out cookies using a 2 in round cookie cutter.
Bake on a parchment-lined baking sheet for 18-20 minutes or until edges are a golden brown and transfer to a cooling rack to cool completely.
Once cooled, whisk together icing ingredients until smooth. Dip half of each cookie in the icing, let excess drip off and run a spoon along the edge to remove any extra, then place on wax paper to set.
For a stronger initial flavor, mix the butter and tea leaves together about 6 hours ahead of time to infuse.
For more, smaller cookies, use a 1 1/2 in round cookie cutter and bake for 12-15 minutes. Will yield around 36 cookies.