Whisk together the coffee, rum and espresso powder in a small bowl until dissolved. Set aside
In the bowl of a stand mixer, whisk the yolks on low until just combined, about 5 seconds. Add the sugar and salt and beat on high until pale yellow, about a minute. Add the mascarpone and coffee liqueur and beat until combined. Scrape the mascarpone mixture into a separate bowl. Pour the cream into the same mixer bowl (no need to clean it) and beat until stiff peaks form, about 1 minute.
Gently, and in batches, fold the whipped cream and mini chocolate chips into the mascarpone mixture. Set aside.
Quickly, and I mean quickly, dunk each side of the lady fingers into the coffee mixture, about 1 second per side. Lay the coffee'd lady fingers in rows on the bottom of a 9-by-13-inch glass baking dish. Once you have a crowded layer, add half of the cream mixture and spread evenly. Repeat another layer of lady fingers, lining them in the opposite direction from the previous layer. Spread on the rest of the cream mixture. Wipe the edges, cover in plastic wrap and let sit overnight, 6 to 8 hours.
Dust in full crust-to-crust coverage with cocoa powder before serving.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.