Preparing the Dough:

1. Warm the milk until just tepid (110°F / 43°C).
2. Combine warm milk, yeast, and 1 tablespoon of sugar in a bowl. Let sit for 5-10 minutes until foamy.
3. In a separate bowl, whisk bread flour, matcha powder, sugar, salt, and dry milk powder.
4. Add the yeast mixture and beaten eggs to the dry ingredients. Mix until just combined.
5. Knead the dough, gradually adding softened butter, for about 8-10 minutes until smooth and elastic.
6. Place the dough in a greased bowl, cover, and let it rise for about 1-1.5 hours until doubled in size.

Making the Cookie Topping:

1. Cream together the butter and powdered sugar until light and fluffy.
2. In a separate bowl, whisk together the flour, matcha powder, and salt.
3. Gradually add the dry ingredients to the butter mixture.
4. Add the egg white and vanilla extract, mixing until a soft dough forms.
5. Wrap the dough in plastic wrap and refrigerate for at least 30 minutes.

Shaping and Assembling:

1. Once the dough has doubled in size, gently punch it down.
2. Divide into 8 equal portions and roll each into a ball.
3. Roll out the cookie topping into 8 thin circles between parchment paper.
4. Drape a circle of cookie dough over each ball of bread dough, then score the tops with a crisscross pattern.
5. Cover loosely and let rise for another 30-40 minutes.

Baking Instructions:

1. Preheat oven to 375°F (190°C).
2. Bake for 12-15 minutes until the cookie topping is lightly golden.
3. Let cool for 5 minutes on the baking sheet, then transfer to a wire rack.
4. Enjoy while still slightly warm for the best texture contrast!