Preheat oven to 350°F and line a baking sheet with parchment paper.
In a large mixing bowl, cream together the butter, brown sugar, and granulated sugar with an electric mixer until creamy. Add in the eggs and vanilla extract and mix again until combined.
In a separate bowl, stir together the flour, cornstarch, matcha powder, salt, baking soda, and baking powder. Add this into the wet mixture and use a silicone spatula to fold together until almost combined. Add in the chopped white chocolate (reserving a handful for later) and continue folding until no dry patches of flour remain.
Using a 2" ice cream scooper, scoop out balls of cookie dough and place on your baking sheet, leaving about 2" between each cookie for spreading.
Bake for about 12 minutes. In the last 2-3 minutes of baking, press a couple of white chocolate chunks into the tops of each cookie and continue baking until the edges are set but the middles are still soft. Let the cookies cool and set for 10 minutes before transferring them to a wire rack to finish cooling.