In a wide, shallow bowl, stir together the coffee, espresso powder, and 3 tablespoons of rum; set aside.
2 cups strong black coffee, 1 tablespoon instant espresso powder, 7 tablespoons dark rum
Fill a medium saucepan (with a diameter a little smaller than the diameter of a heatproof mixing bowl - the mixing bowl should sit just inside the saucepan with at least 1-inch of space between the bottom of the bowl and the water) with 2-inches of water. Bring the water to a low simmer and cover until ready to use.
While the water comes to a simmer, using a hand mixer, whisk the egg yolks in a large heatproof bowl at low speed until just combined. Add the sugar and salt. Beat at medium-high speed until light in color (pale yellow), about 2 minutes. Scrape down the sides of the bowl as needed.
6 large egg yolks, ⅔ cup granulated sugar, ½ teaspoon kosher salt
Add the ⅓ cup of cream to the yolks and beat for 30 seconds at medium speed.
⅓ cup heavy cream
Remove the lid from the saucepan of water and place the mixing bowl with the egg yolk mixture over the simmering water.
Cook, constantly scraping the bottom and sides with a silicone spatula, until the mixture registers 160°F on an instant-read thermometer and coats the back of a spoon, about 5-8 minutes. Watch very closely, especially when the temperature reaches 140°F – the temperature rises quickly. Remove the bowl with the egg mixture from the heat and stir briskly for 1 minute to cool. Set aside to cool to room temperature, stirring periodically, about 20 minutes.
When the egg mixture is at room temperature, whisk in the remaining 4 tablespoons of rum.
Add the mascarpone and cream cheese to the egg mixture. Beat with a hand mixer at medium speed until no lumps remain, about 1 minute. Transfer mixture to a large bowl; set aside.
16 ounces mascarpone cheese, 8 ounces cream cheese
Wipe out the mixing bowl with a paper towel, and attach the whisk to the mixer. Add the ¾ cup heavy cream to the bowl and beat, at medium speed, until frothy, about 1½ minutes. Increase the speed to HIGH and whisk until cream holds stiff peaks, about 1-1½ minutes more.
¾ cup heavy cream
Fold ⅓ of the whipped cream into the mascarpone mixture to loosen, then fold in the remaining whipped cream until completely incorporated; set aside.
Set up a workstation with the ladyfingers, coffee mixture, and 9x13-inch dish all in a row. Working with 1 ladyfinger at a time, quickly roll the ladyfinger into the coffee mixture and place it into the baking dish in a single layer. (Try not to submerge the ladyfinger – just quickly roll it into the coffee mixture – not more than a couple of seconds per ladyfinger.) Trim (or cut with kitchen shears) the ladyfingers to fit, as needed.
14 ounces ladyfingers
Spread half the mascarpone mixture over the single layer of ladyfingers, being sure to cover the sides and corners of the dish. Smooth the top and dust 2 tablespoons cocoa powder over the mascarpone in an even layer (using a small fine-mesh sieve is ideal).
3½ tablespoons cocoa powder
Repeat the process for one more layer and dust with the remaining 1½ tablespoons cocoa powder.
Clean the edges of the dish with a paper towel and cover with plastic wrap. Refrigerate 6-12 hours and sprinkle with grated chocolate, if desired.
¼ cup semisweet chocolate