Preheat oven to 350°F and generously grease a donut baking pan.

In a large bowl, whisk together flour, matcha, baking powder, salt, and sugar.

In another bowl, whisk together 1/3 cup milk, egg, and butter until blended. Pour the egg mixture into the dry ingredients and stir until all the dry bits are just incorporated.

Divide the batter among the donut wells and bake for 12-15 minutes. Let cool in pan briefly, then turn out onto a rack to cool completely.

To make the glaze, whisk together remaining 2 teaspoons milk, powdered sugar and ginger juice until smooth. Use additional sugar or milk to achieve desired consistency. Dip cooled donuts top-side down into glaze or drizzle on with a spoon. Let glaze harden a bit before eating.