For the soaked ladyfingers: Stir coffee, five tablespoons rum, and espresso powder in large bowl until espresso dissolves; set aside. Working with one cookie at a time, quickly dunk ladyfingers in coffee mixture and transfer to wire rack set in rimmed baking sheet; let sit for five minutes.
To assemble: Grease tube pan with vegetable oil spray and line with parchment paper. Lightly coat parchment with oil spray. Line prepared pan with soaked ladyfingers, packing gently to ensure that there are no spaces between cookies; freeze until firm, about one hour. When cookies are firm, scoop ice cream into large bowl and mash with wooden spoon until softened; fold in chocolate chips. Transfer softened ice cream to cookie-lined pan, smooth top, and wrap with plastic wrap. Freeze until ice cream is firm, at least four hours or up to four days.
To serve: With electric mixer on medium speed, beat mascarpone, cream, sugar, and remaining one tablespoon rum until stiff peaks form, about two minutes. Gently invert pan to turn out cake onto serving platter and discard parchment.* Using rubber spatula, top cake with one cup mascarpone mixture and dust lightly and evenly with cocoa. Serve with remaining mascarpone mixture.
*A large tube pan with a removable bottom works best here. If your tube pan doesn't have a removable bottom, it may be necessary to soak the bottom of the pan in a large bowl of hot water for 30 seconds to help the frozen cake release from the pan. Serves 8 to 10
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