Line a 4-cup loaf pan with parchment paper. Melt the white chocolate in a heatproof bowl over simmering water, then remove from the heat.
Whisk the egg yolks and sugar in a bowl until pale and thick, then beat in the melted chocolate.
Heat the cream in a small pan until almost boiling, then remove from the heat. Sprinkle the powdered gelatine over the cream, stirring until completely dissolved. Pour on to the chocolate mixture, whisking vigorously until smooth.
Whisk the yogurt and vanilla extract into the mixture. In a clean, grease-free bowl, whisk the egg whites until stiff, then fold them into the mixture. Pour the mixture into the prepared loaf pan, level the surface and chill until set.
TO MAKE THE SAUCE: Melt the chocolate with the rum and cream in a heatproof bowl over simmering water, stirring occasionally, then leave it to cool completely.
When the mousse is set, remove from the tin. Serve in slices with the cooled chocolate sauce.