Beat the softened butter, light brown sugar, and granulated sugar together in a large bowl with an electric mixer on medium speed until light and creamy, about 3 minutes. Add the egg and vanilla extract and beat until smooth.
In a separate bowl, whisk together the flour, matcha powder, cornstarch, baking soda, and salt until the matcha is evenly distributed with no clumps.
Add the dry ingredients to the wet mixture and stir with a spatula until just combined. Fold in the white chocolate chips. The dough will be a vibrant green color.
Cover the dough and refrigerate for at least 45 minutes until firm enough to scoop.
Preheat your oven to 350°F (177°C) and line baking sheets with parchment paper. Scoop rounded tablespoons of dough and roll into balls. Place them 2 inches apart on the prepared sheets.
Bake for 10 to 12 minutes until the edges are just set but the centers still look slightly soft and puffy. Do not overbake as they firm up as they cool.
Let the cookies cool on the baking sheet for 8 minutes before transferring to a wire rack. Serve once slightly cooled for a soft, chewy center with melty white chocolate pockets.