Line a baking sheet with parchment paper. Set aside.
In a medium bowl, stir together flour, matcha powder, milk powder, salt, baking soda, and baking powder. Set aside.
In a large mixing bowl, use an electric hand mixer or stand mixer to beat the softened butter on medium speed until very creamy. Add the sugar and cream together until fluffy, about 3 minutes. Add in the egg and vanilla extract. Mix again on low speed until combined.
Add the flour mixture to the butter mixture. Mix on low speed until just combined but with some flour patches left. Add the chopped white chocolate, reserving a small handful for later. Finish mixing the dough by folding it together with the chocolate using a spatula until just combined.
Use a 4 tablespoon ice cream scooper to scoop out balls of dough. Place them on your lined baking sheet. No need to space these out. Chill these dough balls in the fridge for at least 1 hour or until firm.
Preheat the oven to 350°F.
Press your thumb into the middle of one ball of dough and press outwards to create a depression. Place a piece of mochi in the middle and press the dough together to wrap around the mochi and form into a ball. Place on your lined baking sheet, leaving at least 2" between each cookie for spreading.
Bake for 12-14 minutes or until the edges are set but the middles are still soft. About 1-2 minutes before they're finished baking, press a few pieces of white chocolate you saved earlier on top of each cookie (optional).
Let the cookies cool on the baking sheet for at least 10-15 minutes before carefully transferring them to a wire rack to cool completely.