For the crust: Preheat oven to 375°. Mix cookie crumbs, sugar, and butter together in a medium bowl, then press mixture onto bottom and up sides of a 9" glass pie plate. Bake until crust is set, 8–10 minutes. Allow crust to cool.
For the filling: Soften gelatin in 1⁄4 cup of the milk in a medium bowl. Finely chop three-quarters of the chocolate and transfer to a small bowl. Scald remaining milk in top of a double boiler set over bottom pot of boiling water, 8–10 minutes. Whisk egg yolks in a large bowl until pale yellow, then whisk in 1⁄2 cup of the sugar, then cornstarch and salt. Gradually whisk in hot milk, then return mixture to top of double boiler and set over bottom pot. Reduce heat to medium-low and cook custard, stirring constantly with a wooden spoon, until thick enough to coat back of spoon, about 15 minutes. Remove custard from heat. Pour 1⁄2 cup of the custard into bowl with chocolate, add vanilla, and stir until smooth. Set chocolate custard aside to cool, 5–10 minutes.
Meanwhile, add remaining custard to bowl with softened gelatin, add rum, and stir well. Cover and refrigerate until cool but not set, 20–25 minutes. Pour chocolate custard into prepared crust and refrigerate until set.
Beat egg whites in a large bowl until medium-stiff peaks form, 1 1⁄2–2 minutes. Gradually add remaining sugar, beating until stiff glossy peaks form, 2–3 minutes. Fold in cooled custard, cover, and refrigerate until just set, 15–20 minutes. Spread custard over layer of chocolate custard and refrigerate over pie.
Whisk cream and confectioners' sugar together in a large bowl until soft peaks form. Spread whipped cream over layer of custard. Shave remaining chocolate over pie.