For the mascarpone filling:

1. Bring a small pot of water to a simmer. Put the egg yolks, sugar, and rum in a stainless steel bowl that will sit on top of the pot. Continuously whisk all three ingredients together until it thickens, about 5 minutes. Cool for 5 minutes.
2. In another bowl, whip the heavy cream with a hand mixer until soft peaks form.
3. Fold the egg mixture to the lightly whipped heavy cream, then add the mascarpone cheese and stir until well combined. Divide the mascarpone cream in two equal parts.
4. Combine matcha powder with boiling water and stir well. Once cooled, add the matcha syrup to half of the mascarpone cream, keeping the other half plain.

For the matcha syrup:

1. Bring water and sugar to a boil. Simmer for 3 minutes and remove from heat.
2. Let cool, then sift in the matcha powder and aged rum. Whisk well.

For assembling the tiramisu:

1. Dunk each ladyfinger into the syrup. Add the drenched ladyfinger to the bottom of a glass, breaking it into 3 pieces. Add one layer of plain mascarpone cream. Sift in some matcha powder over, and place in the freezer for about 5 minutes. Add a second layer of matcha-mascarpone cream, and freeze for 5 minutes.
2. Repeat process once more (soaked lady fingers, mascarpone layer, matcha layer), freezing cream layers for 5 minutes in between. Dust the top of the tiramisu with matcha powder and keep refrigerated until ready to serve, about 2-4 hours.