Stir together the coffee and ½ cup of dark rum in a medium bowl and set aside.

In a large mixing bowl, combine the mascarpone and the remaining ¼ cup of dark rum. Whisk together or beat with a hand mixer on medium speed until well combined. Set aside.

If you have a double-boiler, combine the egg yolks and sugar in the top. Otherwise, combine them in a medium heat-proof mixing bowl. Whisk until well combined. Bring 2 inches of water to a simmer in a small pot or the bottom of a double boiler. Place the egg yolk mixture over the simmering water making sure the water does not touch the bowl. Whisk constantly until the sugar has dissolved and the mixture is light yellow in color and thickened a bit, about 5 to 8 minutes.

Pour the egg yolk mixture into the mascarpone and whisk until combined. Place in the refrigerator for 15 minutes to cool.

In a large mixing bowl or the bowl of a stand mixer fitted with the whisk attachment, combine the cream and vanilla. Beat on medium speed until stiff peaks form, 3 to 5 minutes.

Fold the whipped cream into the mascarpone mixture.

For the Assembly: Working with one at a time, dip the top and bottom of a ladyfinger briefly in the coffee mixture. Arrange enough ladyfingers in a single layer to cover the bottom of a 9x13-inch dish.

Add half of the mascarpone mixture on top of the ladyfingers and smooth out to the edges using an offset spatula. Dip more ladyfingers, arranging them in a single layer on top of the filling as you go. Add the remaining mascarpone mixture and smooth out into an even layer.

Dust top with the cocoa powder and chill overnight.