Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, matcha powder, milk powder, baking powder, baking soda, and salt. Set aside.
In a large mixing bowl, beat the softened butter with an electric mixer until creamy.
Add the granulated sugar and continue beating for 3–4 minutes until light and fluffy.
Add the egg and vanilla extract, mixing until fully combined.
Gradually add the dry ingredients to the wet ingredients and mix until almost combined, leaving a few flour streaks.
Fold in the chopped white chocolate using a silicone spatula until just combined. Do not overmix.
Use a 4-tablespoon cookie scoop to portion the dough onto the prepared baking sheet, spacing cookies about 2 inches apart.
Bake for 12–14 minutes, until the edges are set and the centers remain soft.
During the last 2 minutes of baking, press extra white chocolate pieces onto the tops of the cookies if desired.
Let cookies cool on the baking sheet for 10 minutes before transferring to a wire rack to cool completely.