Preheat the oven to 350°F. Line the bottom of an 8-inch round cake pan (2 inches deep) with parchment; do not grease the sides.
Melt the butter and set aside to cool to room temperature. Sift the cake flour with the salt 2–3 times to aerate; set aside.
In a heatproof bowl, whisk the eggs with the sugar. Set the bowl over a saucepan of barely simmering water (the bowl should not touch the water) and whisk constantly until the mixture is just warm to the touch, 100–110°F, and the sugar is mostly dissolved, 2–3 minutes.
Transfer the warm egg mixture to a stand mixer (or use a hand mixer) and whip on high speed until tripled in volume and very pale; a thick ribbon should fall from the whisk and briefly sit on the surface, 6–8 minutes. Mix in the vanilla during the last 30 seconds.
Using a wide spatula, fold the sifted flour into the foam in three additions, rotating the bowl and scraping to the bottom with each fold until no streaks remain.
Stir about 1 cup of the batter into the melted, cooled butter until smooth, then gently fold this enriched portion back into the main batter just until combined.
Immediately pour the batter into the prepared pan and smooth the top. Bake until the cake is golden, the center springs back when lightly pressed, and a toothpick comes out clean, 22–28 minutes.
Cool in the pan on a rack for 10 minutes. Run a thin knife around the edge, invert onto the rack, peel off the parchment, then turn right-side up. Cool completely until the crumb is set, 45–60 minutes, before slicing or filling.
Serve plain, dusted with sugar, or use as a base for layered cakes. Wrap leftovers tightly and store at room temperature for up to 2 days.