Whisk together the espresso powder and 1/4 cup hot water in a medium bowl until the espresso powder completely dissolves. Add 3/4 cup of the dulce de leche and stir until well combined.
Beat 3 cups of the heavy cream in a large bowl with an electric mixer on medium speed to medium peaks, about 2 minutes. Add the espresso and dulce de leche mixture to the cream and beat to very stiff peaks, about 1 minute more.
Arrange the chocolate wafer cookies to make a 9 1/2-by-4-inch rectangle directly in the center of a large, flat serving platter (breaking up 1 of the cookies to fill in any gaps). Spread 1/4 cup of the espresso-dulce de leche cream on top of the cookies. Lay 8 ladyfingers flat across this cream layer and then spread a third of the remaining cream mixture on top of the ladyfingers. Repeat this process 2 more times with the remaining ladyfingers and cream mixture. Use an offset spatula to spread the top layer of cream mixture evenly over the top and down the sides to cover the icebox cake completely. Cover with plastic wrap and refrigerate 8 hours and up to overnight.
Once completely chilled, remove the plastic wrap from the cake. Beat the remaining 1/2 cup heavy cream in a medium bowl with an electric mixer (or by hand) to medium peaks. Spread the whipped cream on top of the cake. Microwave the remaining dulce de leche in a small microwave-safe bowl until it is loose enough to drizzle. Drizzle evenly on top of the cake, then sprinkle with the crushed espresso beans.
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