Melt butter and allow to cool. Set aside.
In a large bowl, use an electric hand mixer to beat eggs and egg yolk until light in color. Slowly add sugar and continue to beat until mixture is thick and pale in color, about 3-5 minutes. Stir in vanilla.
In another bowl, whisk together flour, matcha powder, baking powder, and salt. Use a spatula to gently fold dry ingredients into egg mixture.
Add cooled melted butter in two additions. Fold until butter is absorbed into batter. Transfer batter to an airtight container. Let chill in the fridge for at least three hours, preferably overnight.
Preheat oven to 375°F. Butter and flour a 12 cavity madeleine pan. Tap out any excess flour.
Fill madeleine pan ¾ full of batter. Use a piping bag, small cookie scoop, or two spoons to distribute the batter.
Bake for 10-12 minutes until cakes have a little hump and the top of cake springs back when gently pressed. Allow cakes to cool in pan for one minute before inverting and unmolding. Allow cakes to cool slightly before dusting with powdered sugar.
Eat as soon as madeleines are cool enough to pick up. Madeleines are best enjoyed the same day they are baked.