Preheat the oven to 350°F (175°C). Line two large baking sheets with parchment paper.
In a large bowl, beat the softened butter and granulated sugar with an electric mixer on medium speed until smooth and creamy, about 1 to 2 minutes. Beat in the egg and vanilla until fully combined.
In a separate bowl, whisk together the flour, matcha powder, baking powder, and salt, breaking up any visible matcha lumps.
Add the dry ingredients to the butter mixture. Mix on low speed just until a soft dough forms and no dry streaks remain. Do not overmix.
Scoop level tablespoons of dough onto the prepared baking sheets, leaving about 2 inches (5 cm) between portions. Roll each portion into a ball, then gently flatten it to about 1/2 inch (1 cm) thick. Sprinkle with coarse sugar, if using.
Bake one sheet at a time for 9 to 11 minutes, until the edges are set and the bottoms are very lightly golden. The tops should remain green rather than brown.
Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely. They will firm up as they cool.