Preheat oven and line pan:
Set oven to 350°F (175°C). Line an 8x4x2-inch loaf pan with parchment; reserve.Whisk dry ingredients:
In a large bowl, whisk together 1 cup (125g) flour, sifted matcha, sugar, baking powder, baking soda, and salt.Mix wet and dry ingredients:
Add mashed banana, melted butter, and milk to the dry mix. Beat on low until combined, then on high for 2 minutes. Add eggs and remaining ¾ cup (93g) flour; beat until smooth. Fold in white chocolate chips and walnuts using a rubber spatula until evenly dispersed.Bake and cool loaf:
Transfer batter to the prepared pan. Bake 55–60 minutes until a toothpick inserted in the center emerges clean. Cool on a wire rack for 10 minutes, then unmold.