Step 1: Combine the Dry Ingredients
In a medium-sized bowl, place a fine-mesh sieve over the top. Add the all-purpose flour, matcha powder, baking soda, and salt. Sift all the ingredients together into the bowl. Sifting is especially crucial for the matcha powder, which tends to clump. This step ensures an even distribution of flavor, a uniform green color, and a smooth texture in the final cookie. Whisk the sifted ingredients together to fully combine them and set the bowl aside.

Step 2: Mix the Wet Ingredients
In a separate large bowl, add the melted and slightly cooled unsalted butter. Add the packed light brown sugar and granulated sugar to the butter. Whisk vigorously for about a minute until the mixture is smooth, glossy, and well combined. The sugar should begin to dissolve into the butter. Add the room-temperature egg, the extra egg yolk, and the vanilla extract. Whisk again until the mixture is pale, thick, and fully emulsified.

Step 3: Combine Wet and Dry
Pour the dry ingredient mixture into the large bowl with the wet ingredients. Using a rubber spatula or a wooden spoon, gently fold the ingredients together until they are just combined. It is very important not to overmix at this stage. Stop mixing when you still see a few small streaks of flour. Overmixing will develop the gluten in the flour and result in tough, dense cookies.

Step 4: Fold in the White Chocolate
Add the chopped white chocolate or callets to the dough. Fold them in gently until they are evenly distributed throughout the dough. At this point, the dough will be quite soft and sticky.

Step 5: The Crucial Chilling Step
This is the most important step for cookie success. Cover the bowl with plastic wrap and transfer it to the refrigerator. Chill the dough for a minimum of 2 hours, but for the absolute best flavor and texture, chill it for 24 to 48 hours. This mandatory rest period does two things:

1. Solidifies the Fat: Chilling the butter prevents the cookies from spreading into thin, greasy crisps in the oven. It ensures they bake up thick and chewy.

2. Hydrates the Flour & Matures the Flavor: The rest time allows the flour to fully absorb the liquid, and the flavors of the matcha, vanilla, and butter to meld and deepen significantly. A well-rested dough produces a cookie with a much more complex and delicious flavor profile.

Step 6: Prepare for Baking
When you are ready to bake, position your oven racks in the upper and lower thirds of the oven and preheat to 350°F (175°C). Line two large baking sheets with parchment paper.

Step 7: Scoop and Bake
Remove the chilled dough from the refrigerator. It will be firm. Using a medium cookie scoop (about 1.5 to 2 tablespoons), scoop out balls of dough and place them on the prepared baking sheets, leaving at least 2 inches of space between each cookie to allow for spreading. For a bakery-style look, you can press a few extra pieces of chopped white chocolate onto the tops of the dough balls before baking.

Bake for 11-13 minutes, rotating the baking sheets halfway through the baking time (swapping their positions on the racks and turning them 180 degrees) to ensure even baking. The cookies are done when the edges are set and lightly golden brown, but the centers still look slightly soft and underdone. Do not overbake them, as they will continue to set on the hot pan after being removed from the oven.

Step 8: Cool for Optimal Chewiness
Remove the baking sheets from the oven. For an extra chewy, wrinkly texture, you can carefully lift the baking sheet a few inches and drop it flat onto the counter once or twice (this is optional but effective). Let the cookies cool on the baking sheet for 5-10 minutes. This allows them to firm up enough to be moved. After 10 minutes, transfer the cookies to a wire rack to cool completely.