Use this recipe to frost the outer layer of the Birch de Noel cake. The simplest way to test whether the sugar has dissolved is by rubbing the mixture between your fingers; it should feel completely smooth with no remaining granules.
In a small, heavy saucepan, combine 3/4 cup sugar, 1/3 cup water, and corn syrup; cook over medium heat, stirring occasionally, until sugar dissolves. Increase heat to high and bring to a boil, washing down sides of pan with a pastry brush dipped in water (to stop crystals from forming). Cook, without stirring, until a candy thermometer registers 230°, about 5 minutes.
Meanwhile, in a mixer fitted with the whisk attachment, beat egg whites on medium speed until soft peaks form, about 2 1/2 minutes. Gradually add remaining 1 tablespoon sugar until combined.
Once sugar syrup has reached 230°, immediately remove from heat. With mixer on medium-low, carefully pour syrup into egg-white mixture, in a steady stream down sides of the bowl (to prevent splattering). Increase speed to medium-high and beat until mixture is completely cool and stiff peaks form, about 7 minutes. Add vanilla and beat until fully incorporated and frosting is shiny and smooth. Use immediately.