In a large bowl, whip the heavy cream, pudding mix, and vanilla until thick and billowy, with firm peaks.
Briefly dip the biscuits in the milk. Arrange a layer in a 9-by-5-inch (23-by-13-cm) loaf pan and spread with half the cream. Top with another layer of biscuits and the remaining cream, smoothing the surface.
Freeze until firm, about 3 hours.
Combine the dark chocolate, white chocolate, and oil in a microwave-safe bowl. Microwave in 20-second intervals, stirring between each, until melted and smooth.
Pour the chocolate mixture over the frozen cake, spreading it into an even layer. Sprinkle lightly with flaky sea salt.
Return to the freezer until the topping is set, about 20 minutes.
Let stand at room temperature for 30 minutes before serving.