Preheat oven to 350°F. Line 9x5-inch loaf pan with parchment paper, along paper overhang to extend along the two long sides. Generously grease lined pan with butter. Set aside.
Sift matcha powder into a medium bowl. Pour in hot milk. Whisk until mixture is smooth and matcha powder has dissolved. Set aside and allow to cool.
In the bowl of a stand mixer fitted with a paddle attachment (or using an electric hand mixer), cream butter and sugar on medium speed until smooth and pale, about 3 minutes.
Mixing on low speed, add eggs one at a time until thoroughly incorporated. Add vanilla. Scrape down sides of the bowl as needed to ensure thorough mixing.
In a large bowl, sift together all-purpose flour, cake flour, salt, baking powder, and baking soda. On low speed, add dry mixture to wet batter in three additions, alternating with the sour cream. Mix just until there are no longer any dry streaks of flour.
Divide batter in half. Add dissolved matcha to one bowl. Fold until matcha has been throughly incorporated into batter.
Alternate adding dollops of vanilla batter and matcha batter into the prepared loaf pan. When finished, use a mini offset spatula or the back of a spoon to smooth the top of the batter into an even layer.
Bake for about 45-55 minutes until toothpick inserted in center of cake comes out clean. Cool in pan for 10 minutes. Unmold loaf and continue to cool to room temperature on wire rack. Slice room temperature cake with a serrated knife.