Cake

• Vegetable-oil cooking spray
• ½ cup sifted unsweetened Dutch-process cocoa powder, plus more for dusting
• 1 cup sifted cake flour (not self-rising)
• ½ teaspoon kosher salt
• 6 large eggs, plus 4 large egg yolks, room temperature
• 1 cup sugar
• 1 stick unsalted butter, melted and cooled

Filling

• 2 ½ teaspoons unflavored gelatin (one ¼-ounce package)
• ⅓ cup light rum
• 4 large eggs, separated, room temperature
• 1 ¼ cup sugar
• 2 ½ cups cold heavy cream
• ½ teaspoon freshly grated nutmeg
• 2 teaspoons pure vanilla extract

Frosting

• 5 large egg whites, room temperature
• 1 ¼ cups sugar
• ¼ teaspoon cream of tartar
• ¾ teaspoon pure vanilla extract