In a medium-sized mixing bowl or bowl of a stand mixer, mix together sugar, cake flour, baking powder, baking soda, salt, matcha powder, and ginger.
Add butter and mix on medium-low speed for three minutes. Because there is so little butter, you'll end up with a very fine crumb texture.
In a small mixing bowl, whisk together eggs, yogurt, and oil until smooth.
Add the egg mixture to the flour mixture and beat on medium speed until just combined.
Slowly add milk and mix on low speed until just combined. The batter will be liquid. (Don't worry, you didn't do anything wrong. It's supposed to be that way.)
Fill cupcake liners just over 1/2 full.
Bake for 14 minutes and then test to see if they are done. They are done when a toothpick comes out without wet batter stuck to it. If they are not done, test again in two minutes. If they are still not done, test again in another two minutes.
When the cupcakes are done, remove them immediately from the tins and leave them on a cooling rack (or just on your counter if you don't own a cooling rack) to cool.
Mix butter on high speed for three minutes or until light and fluffy.
Mix in red bean paste until smooth.
Mix in salt.
Mix in powdered sugar a little bit at a time until fully combined.