Preheat the oven to 350 degrees F. Center a rack in the oven. Butter a 9-inch round cake pan. Dust with flour, tapping out the excess.
Pour a few inches of water into a skillet large enough to hold the bowl from your mixer. Bring the water to a gentle simmer. Melt the butter and set it aside to cool. It should be just warm when you're ready for it.
Whisk the eggs and sugar together in the mixer bowl. Place the bowl in the skillet of simmering water and, whisking without stop, heat the mixture until it is foamy, slightly pale, and reads between 130 and 140 degrees F on an instant-read thermometer, about 4 minutes. Remove the bowl from the water. Working in the mixer fitted with the whisk attachment, beat the mixture on high speed until it cools to room temperature and triples in volume. When you lift the whisk, the batter will fall back into the bowl and form a ribbon that remains on the surface for 10 seconds before it dissolves, about 5 to 8 minutes.
Stir about 2 tablespoons of batter into the slightly cooled butter and set it aside. With a large flexible rubber spatula, gently fold the flour into the batter in 2 or 3 additions, taking care to handle the batter gently in order to maintain its bubble structure. (The batter will lose volume as you fold in the flour and later the butter. This reaction is inevitable and shouldn't jeopardize the success of the finished cake, but be careful just the same.) Still working with the spatula, fold in the butter mixture.
Pour the batter into the prepared pan and bake until the top is golden and springy to the touch and a toothpick inserted in the center of the cakes comes out clean, 28 to 33 minutes. Transfer the cake to a cooling rack for 5 minutes. Remove cake from pan, invert, and cool completely on the rack.
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