Combine chocolate, rum, 1/4 cup water, and the vanilla in a heat-proof bowl. Set it over a pan of simmering water just until the chocolate melts. Cool completely to room temperature. Whisk butter until combined. Set aside.
Place egg whites, the salt, and 1 tablespoon of the sugar in a bowl. Beat on high speed until firm but not dry.
Place the egg yolks and the rest of the sugar in a bowl. Beat on high speed until pale yellow. With the mixer on low speed, add the chocolate mixture.
Whisk 1/4 of the egg whites into the chocolate mixture; then fold the rest in carefully with a rubber spatula.
Fold the whipped cream into the chocolate mixture.
Soak the biscuits in milk slightly. Layer a bottom of a square pan with them. Next layer with chocolate mixture (about 1/2 inch high). Sprinkle with cashew nuts. Follow with another layer of soaked biscuits and then the chocolate mixture and cashew nuts. Continue layering until all of the chocolate mixture is over. Make sure that the top layer is chocolate mixture and cashew nuts.