Adapted for high-altitude from Rose Levy Beranbaum recipe.
With some extra butter, thoroughly grease an 8 inch springform. Cover it with some extra flour, roll it around, and tap out the excess.
Melt butter over medium heat. Let the butter boil butter until the milk solids fall to the bottom and begin to brown. You want them to be a deep golden brown color with a nutty smell. Pour through a fine mesh strainer, and allow to come to room temperature.
In a large mixing bowl set over a pan of simmering water, combine the eggs and sugar and hand whisk until just lukewarm. Place bowl on stand mixer fitted with whip attachment and whip it until very thick and tripled in volume. This will take about 5 minutes. While the whipping is happening, sift the flour and cornstarch.
Remove about 3/4 cup egg mixture and whisk in the warm and liquid (not scalding) beurre noisette. Add in the vanilla.
Sift half the flour mixture over the remaining eggs and fold in gently but thoroughly. You’re trying to minimize loss of volume here, so be gentle (a slotted skimmer or balloon whip works well). Repeat with rest of flour. Turn until flour just combined, then fold in the butter mixture.
Pour batter into prepared pan, smooth the top, and bake at 370 F without opening the oven, until the cake is golden brown and the cake starts to shrink away from the sides of the pan a bit. This should take between 20 and 30 minutes. No need to jab a skewer into it: shrinkage=done.
Immediately loosen cake from sides of pan and turn it out. For cooling and serving, top must be upright to prevent deflation. If not used immediately, brush top and bottom with simple syrup.