In a large bowl, beat butter with sugar until light and fluffy; beat in egg and vanilla. Whisk together flour, matcha green tea powder, baking powder and salt; gradually stir into butter mixture until dough comes together.
Divide dough in half and wrap each in plastic wrap. Refrigerate for 1 hour. Let stand at room temperature until soft enough to roll out, about 15 minutes.
Preheat oven to 350 °F. Line baking sheet with parchment paper. Set aside.
Place dough on a lightly floured surface and roll to ¼ inch thick. Use cookie cutter to make cookie shapes, re-rolling any scraps. Repeat process with the other dough.
Place cookies on prepared baking sheet and bake for 10 to 12 minutes or until edges are light brown. Let cookies cool for a minute on baking sheets on racks then transfer to racks and let cool completely.