To make the cake, position a rack in the center of the oven and heat to 350°F. Butter a 9-inch springform pan, and line the bottom with a round of parchment or wax paper. Sprinkle the inside of the pan with flour, coating evenly and tapping out the excess flour.
In a small saucepan, bring the butter to a boil over medium heat. Cook for about 1 minute, just until the milk solids in the butter turn light brown. Remove from the heat and let stand for 5 minutes. Skim the foam from the top, then pour into a medium bowl, leaving the browned bits in the pan. Stir in the vanilla.
In a large mixing bowl, using a hand-held electric mixer on high speed, beat the yolks with ⅓cup of the sugar until the mixture is very pale and forms a thick ribbon that falls back on itself when the beaters are lifted a few inches, about 3 minutes.
In another large bowl, using clean beaters, beat the whites on high speed until they form soft peaks. Gradually beat in the remaining ⅓cup of the sugar until the peaks are shiny and stiff.
Stir a large dollop of the whites into the yolks to lighten the mixture, then fold in the remaining whites. Combine the flour and salt. In two additions, sift the flour over the eggs, and use a large balloon whisk or rubber spatula to fold together.
Whisk a large dollop of the batter into the melted butter. Fold this mixture back into the batter. Pour into the pan and smooth the top.
Bake until the top springs back when pressed in the center and the sides are barely beginning to shrink from the pan, about 30 minutes. Cool on a wire rack for 5 minutes. Remove the sides of the pan, invert onto the rack, and remove the bottom of the pan and the paper. Reinvert the cake onto another rack, right side up. Cool completely.