For the crust: Preheat the oven to 350 degrees F. Process the graham crackers, brown sugar and ginger in a food processor until they form a fine crumb. Add the butter and rum, and pulse until the crumbs are soft and moist. Press the crumbs into the bottom and up the side of a 9-inch pie plate, and bake until firm, 12 to 15 minutes. Let the crust cool completely on a wire rack.
For the rum pudding: Set a fine sieve over a mixing bowl; set aside. Whisk the milk, brown sugar and cornstarch in a medium saucepan until the cornstarch is dissolved and no lumps remain. Add the egg yolks, ginger and salt, and whisk until combined. Bring the mixture to a boil over medium-low heat, whisking constantly, 8 to 10 minutes. When the mixture is thick, reduce the heat to low, and cook for 1 minute more, whisking vigorously. Pour the pudding through the sieve set over the mixing bowl, and use a spatula to press it through (this will catch any eggy bits). Whisk the butter into the pudding until it's incorporated, then whisk in the rum. Pour the pudding into the cooled pie shell. Lay plastic wrap directly on the pudding, and refrigerate the pie until it's cool and set with a slight jiggle, about 2 hours.
For the rum-and-lime whipped cream: Beat the confectioners' sugar, rum, lime juice and salt in a large bowl with an electric mixer on medium-high speed until combined. Add the heavy cream, and beat until fluffy peaks form.
Transfer the whipped cream to a pastry bag fitted with a star tip. Pipe the whipped cream onto the cooled pie, and garnish with crystallized ginger. The pie can be refrigerated, loosely covered, up to overnight or served immediately.
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