Preheat the oven to 350 degrees Fahrenheit. Grease and flour a nine- by five-inch loaf pan. Mash the bananas in a medium bowl with a potato masher. Set aside.
In a large mixing bowl, beat the butter or margarine and sugar with a stand or hand-held electric mixer on medium-high speed, or whisk by hand until creamy. Scrape down the bowl a few times.
Add the eggs and mix well on high speed, scraping down the sides and bottom of the bowl a few times. Add the mashed bananas and mix well.
Add the flour, baking powder, baking soda, and salt and mix on low speed to combine. Add the 1/4 cup milk and vanilla and stir gently until combined.
Scoop up half of the batter, about one and 1/4 cups, into another bowl and set aside.
To the mixer bowl, add the matcha green tea powder and remaining one tablespoon milk and whisk in well.
You will need two silicone spatulas and two 1/4-cup measuring cups. Scoop up a heaping 1/4 cup of the green batter and dump into the loaf pan and spread with a spatula to cover the bottom. The easiest way to do this is to use the tip of the spatula to spread the batter towards the sides of the pan, turning the pan around to spread on all sides and cover as best as you can. Scoop up a heaping 1/4 cup of the white batter and dump on top in about six to eight clumps in a few places on top of the green. Use the spatula to gently spread over the green batter, being careful not to press down too hard to mix the batters. Repeat for three more layers of each batter, one at a time, alternating colors, for a total of four layers of each batter.
Bake for 50 minutes, or until a skewer inserted in the center comes out clean. Let cool in the pan for 10 minutes and then remove to a cooling rack and let cool.
To make the icing, place the confectioners’ sugar into a bowl. Add the matcha green tea powder and mix well. Add the hot water, teaspoons at a time, until you have a thick icing.
When the cake is mostly cool, spread the icing over the top of the cake. If the icing gets too thick, add a bit more water; if too thin, add more confectioners’ sugar.
Store covered at room temperature for up to five days or freeze for up to three months.