Make the matcha syrup first by bring water (3/4 cup) and sugar (1/2 cup) to a boil. Simmer for 3 minutes and remove from heat.
Let cool, then sift in the matcha (1 tablespoon). Whisk well. (This can be made up to a day before.)
Matcha Sponge Cake
Turn oven on to 350°F. Lay parchment paper on a 13"x18" sheet tray. If the parchment paper rolls up, put a little butter or oil under each corner. Dust the parchment paper with confectioner's sugar. Set aside.
Separate eggs (4). In a mixing bowl, whisk on low speed, the egg whites for 1 minute. Turn up the speed to medium and slowly add the sugar (7 tablespoons).
After adding the sugar, turn speed to high. Mix until the stiff peaks form and the egg whites look shiny. Add the egg yolks (4) and fold to mix. Sift in matcha (1/2 tablespoon) and flour (3/4 cup) to the egg mixture and fold to incorporate.
Pour the mixture onto the sheet tray and spread evenly. Dust confectioner's sugar on top.
Bake for 10 minutes. Use a cookie cutter (we used the 3") to cut out circles.
Bring a small pot of water to a simmer. Put the egg yolks (2), sugar (1 tablespoon), and rum (1 tablespoon) in a stainless steel bowl that will sit on top of the pot.
Continuously whisk all three ingredients together until it thickens to the "ribbon stage" (about 5 minutes). Cool for 5 minutes.
In another bowl, whip the heavy cream (1/2 cup) until soft peaks form. Combine the lightly whipped heavy cream and the egg mixture. Fold together.
Sift in the matcha (1 tablespoon) and mix well. Finally, fold in the mascarpone cheese (2/3 cup).
Assembling the Matchamisu
In a small glass (we used mini cups), put in a cut out round piece of the sponge cake. (We cut a smaller piece of cake to fit the tapered bottom of the cup.)
Spoon on the matcha syrup over the sponge cake, making sure the sponge cake gets soaked with the syrup. Dollop the matcha cream on top.
Layer on another piece of sponge cake. Pour on matcha syrup to the top level sponge cake. Add the final layer of matcha cream and dust with matcha.