Make the cake: Heat oven to 325°. Line bottom of 9" springform pan with parchment paper. Grease pan with 1 tbsp. butter. Add 1⁄4 cup flour; shake to evenly coat inside of mold. Invert and tap out excess flour; set pan aside.
In a large bowl, beat egg yolks and 1⁄4 cup sugar with a handheld mixer on medium speed until pale and airy, 4–5 minutes; set yolk mixture aside. In a large nonreactive bowl, beat egg whites on high speed until frothy. Sprinkle in remaining sugar; beat to form stiff, glossy peaks. Beat one-fourth of the egg whites into yolk mixture. Using a rubber spatula, fold in remaining egg whites. Fold in melted butter, remaining flour, and salt. Pour batter into prepared pan; smooth top. Bake until a toothpick inserted in cake comes out clean, about 1 hour. Transfer to a rack; let cool completely.
Make the filling: Combine gelatin and 1⁄4 cup cold water in a small bowl and set aside. In a large metal bowl, whisk together 8 egg yolks, 1⁄2 cup sugar, and 1⁄3 cup cornstarch; set aside. In a 2-qt. saucepan, bring milk, cream, and reserved vanilla seeds to a boil. While whisking constantly, slowly drizzle milk mixture into yolk mixture. Place bowl over a 2-qt. saucepan of simmering water. Cook, whisking constantly, until mixture thickens, 4–5 minutes. Remove from the heat and whisk in gelatin mixture; let cool to room temperature. Cover surface of filling with plastic wrap and refrigerate until set, about 4 hours.
Invert cooled cake onto work surface and slice into 3 horizontal layers (see How-To: Cutting Cake Layers). Place the bottom cake layer on a cardboard cake circle. Using a pastry brush, brush cake with one-third of the rum. Using a butter knife, spread half of the filling over cake. Top with second cake layer, brush with one-third of the rum, and spread with remaining filling. Top with remaining cake layer and brush with remaining rum; set cake aside.
Make the meringue frosting: Heat 2 cups sugar and 1⁄2 cup water in a 2-qt. saucepan over medium-high heat to make a syrup. Cook, without stirring, until a candy thermometer inserted in the syrup reaches 240°. Meanwhile, in bowl of a standing mixer fitted with a whisk, beat confectioners' sugar, 1 1⁄2 tsp. cornstarch, cream of tartar, and 4 egg whites on medium speed until soft peaks begin to form, about 1 minute. While continuing to whisk, add the hot sugar syrup in a slow, steady stream to the egg whites. On high speed, beat until doubled in volume to make a thick and smooth frosting, about 10 minutes.
Using a butter knife, spread 1 1⁄2 cups of the frosting evenly over top and sides of cake in a 1⁄4"-thick layer. Transfer remaining frosting to a pastry bag fitted with a 1⁄4"-diameter tip. Pipe top and sides decoratively with frosting. (Alternatively, spread remaining frosting over cake in swirls.) To lightly brown frosting (an optional step), light a small culinary butane torch. Hold torch 4" from cake and lightly caramelize frosting. Decorate with candied violets and sprinkle with pistachios.