Preheat oven to 325 degrees F. Butter the bottom and sides of a 9-inch springform pan that is at least 3 inches high.

Melt the butter, chocolate, and espresso powder in a large heatproof bowl set over a saucepan filled with 1 inch of barely simmering water, stirring occasionally until smooth. Remove from heat and cool for 5 minutes. Whisk in the vanilla and egg yolks; set aside.

In a stand mixer fitted with the whisk attachment, beat the egg whites and salt at medium speed until frothy, about 30 seconds. Add half of the brown sugar and beat until combined, about 15 seconds. Add the remaining brown sugar and beat at high speed until soft peaks form when the whisk is lifted, about 1 minute longer, scraping down the sides halfway through.

Whisk one-third of the beaten egg whites into the chocolate mixture. Using a rubber spatula, fold in the remaining egg whites until no white streaks remain. Carefully transfer the batter to the prepared springform pan, gently smoothing the top with an offset spatula.

Bake until the cake has risen, is firm around the edges, and the center has just set but is still soft, 13 to 18 minutes. Transfer the pan to a wire rack to cool completely, about 1 hour. (The cake will collapse as it cools.) Do not remove the cake from the pan.

Whisk together the cocoa powder and hot water in a small bowl and set aside. Melt the chocolate in a large heatproof bowl set over a saucepan filled with 1 inch of barely simmering water, stirring occasionally until smooth. Remove from heat and cool for 5 minutes.

In the bowl of a stand mixer fitted with the whisk attachment, whip the cream, granulated sugar, and salt on medium speed until the mixture begins to thicken, about 30 seconds. Increase the speed to high and whip until soft peaks form when the whisk is lifted, 15 to 60 seconds.

Whisk the cocoa powder mixture into the melted chocolate until smooth. Whisk one-third of the whipped cream into the chocolate mixture to lighten. Using a rubber spatula, fold in the remaining whipped cream until no white streaks remain.

Spoon the mousse into the springform pan over the cooled cake and gently tap the pan on the counter 3 times to remove any large air bubbles; smooth the top with an offset spatula. Refrigerate for at least 15 minutes while preparing the top layer.

In a small bowl, sprinkle the gelatin over the water; let stand for at least 5 minutes. Place the white chocolate in a medium bowl. Bring ½ cup of the heavy cream to a simmer in a small saucepan over medium-high heat.

Remove from heat and add the gelatin mixture, whisking until fully dissolved. Pour the cream mixture over the white chocolate chips and whisk until the chocolate is melted and the mixture is completely smooth. Cool to room temperature, stirring occasionally, 5 to 8 minutes.

In the bowl of a stand mixer fitted with the whisk attachment, whip the remaining 1 cup heavy cream at medium speed until it begins to thicken, about 30 seconds. Increase the speed to high and whip until soft peaks form when the whisk is lifted, 15 to 60 seconds.

Whisk one-third of the whipped cream into the white chocolate mixture to lighten. Using a rubber spatula, fold the remaining whipped cream into the white chocolate mixture until no white streaks remain.

Spoon the white chocolate mousse into the pan over the middle layer. Smooth the top with an offset spatula. Return the cake to the refrigerator and chill until set, at least 2½ hours.

Remove the cake from the refrigerator and let sit at room temperature for 45 minutes before releasing it from the pan. Run a thin knife between the cake and side of the springform pan, then remove the side of the pan. Cut into slices and serve.