1. For the bottom layer; Adjust oven rack to middle position and heat oven to 325 degrees. Butter bottom and sides of 9-1/2-inch springform pan. Melt butter, chocolate, and espresso powder in large heatproof bowl set over saucepan filled with 1-inch of barely simmering water, stirring occasionally until smooth. Remove from heat and cool mixture slightly, about 5 minutes. Whisk in vanilla and egg yolks; set aside.
2. In stand mixer fitted with whisk attachment, beat egg whites and salt at medium speed until frothy, about 30 seconds. Add half of brown sugar and beat until combined, about 15 seconds. Add remaining brown sugar and beat at high speed until soft peaks form when whisk is lifted, about 1 minute longer, scraping down sides halfway through. Using whisk, fold one-third of beaten egg whites into chocolate mixture to lighten. Using rubber spatula, fold in remaining egg whites until no white streaks remain. Carefully transfer batter to prepared springform pan, gently smoothing top with offset spatula.
3. Bake until cake has risen, is firm around edges, and center has just set but is still soft (center of cake will spring back after pressing gently with finger), 13 to 18 minutes. Transfer cake to wire rack to cool completely, about 1 hour. (Cake will collapse as it cools.) Do not remove cake from pan.
4. For the middle layer: Combine cocoa powder and hot water in small bowl; set aside. Melt chocolate in large heatprooof bowl set over saucepan filled with 1 inch of barely simmering water, stirring occasionally until smooth. Remove from heat and cool slightly, 2 to 5 minutes.
5. In clean bowl of stand mixer fitted with whisk attachment, whip cream, granulated sugar, and salt at medium speed until mixture begins to thicken, about 30 seconds. Increase speed to high and whip until soft peaks form when whisk is lifted, 15 to 60 seconds.
6. Whisk cocoa powder mixture into melted chocolate until smooth. Using whisk, fold one-third of whipped cream into chocolate mixture to lighten. Using rubber spatula, fold in remaining whipped cream until no white streaks remain. Spoon mousse into springform pan over cooled cake and gently tap pan on counter 3 times to remove any large air bubbles; gently smooth top with offset spatula. Wipe inside edge of pan with damp cloth to remove any drips. Refrigerate cake at least 15 minutes while preparing top layer.
7. For the top layer: In small bowl, sprinkle gelatin over water; let stand at least 5 minutes. Place white chocolate in medium bowl. Bring 1/2 cup cream to simmer in small saucepan over medium-high heat. Remove from heat; add gelatin mixture and stir until fully dissolved. Pour cream mixture over white chocolate and whisk until chocolate is melted and mixture is smooth, about 30 seconds. Cool to room temperature, stirring occasionally, 5 to 8 minutes (mixture will thicken slightly).
8. In clean bowl of stand mixer fitted with whisk attachment, whip remaining cup cream at medium