Mix coffee, milk, 2 tablespoons sugar, and 2 tablespoons rum together in a bowl. Soak lady fingers in the coffee mixture. Layer 1/2 of the soaked ladyfingers in the bottom of a 9x13 inch pan.
Beat egg yolks and 2 tablespoons sugar together in a bowl; add mascarpone cheese and 1 tablespoon of rum and mix well.
Beat egg whites in a bowl using an electric mixer until stiff peaks form; fold into mascarpone cheese mixture. Spread 1/2 of mixture over soaked lady fingers; cover with second 1/2 of soaked lady fingers and cover evenly with remaining mascarpone cheese mixture. Sprinkle top with cocoa powder. Refrigerate at least 3 hours.