Preheat oven to 325 degrees. Grease a 10-inch springform pan with non-stick cooking spray. Place chocolate and 3 tablespoons of water in a heatproof bowl set over a pan of simmering water. Stir until melted and smooth; set aside to cool slightly.
Place egg yolks and sugar in the bowl of an electric mixer fitted with the whisk attachment. Beat on high speed until pale and thick, about 3 minutes. On the medium-low setting, add vanilla and rum. Switch to paddle attachment. On medium speed, gradually add the chocolate mixture and the butter, mixing them together until smooth.
Transfer mixture to a large bowl. Sift flour, cocoa, and salt over top of mixture, then fold in; set aside.
Place the egg whites in a clean bowl of an electric mixer fitted with the whisk attachment. Beat the whites until almost-stiff peaks form. Fold them into the chocolate mixture in two additions (being careful not to overmix). Carefully transfer the batter to the pan and bake until puffed, cracked, and set, about 1 hour and 20 minutes.
Remove from oven and let cool completely in pan on wire rack. Just before serving, remove the cake from the pan and dust with powdered sugar.