Combine cake mix and 2 tablespoons of the instant espresso powder. Prepare and bake the cake according to the package directions for a 9-by-13-inch baking dish. Cool completely on a wire rack. Use the handle of a wooden spoon to poke 25 holes in the cake (make 5 rows of 5).
Whisk together the sweetened condensed milk, pudding mix, 1/2 cup of the heavy cream and 1/2 cup liqueur in a medium bowl for 2 minutes. Let sit for 5 minutes for the pudding to set.
Meanwhile, stir together the remaining 2 tablespoons liqueur and remaining 1 tablespoon espresso powder in a small bowl until the powder dissolves. Whip the remaining 1 1/2 cups heavy cream together with the confectioners' sugar in a large bowl with an electric mixer on medium speed until stiff peaks form, about 3 minutes. Add the espresso liqueur mixture and whip just to incorporate.
Spread the pudding mixture over the top of the cake, making sure the holes get filled completely. Spread the whipped cream evenly over the surface. Chill at least 1 hour or up to overnight.
About 30 minutes before serving, melt the chocolate chips in a microwave in 30-second increments, stirring between each, until soft. Let cool slightly. Drizzle over the top of the cake, then sprinkle with the chopped espresso beans and serve cold.
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