In a heavy skillet, melt the butter over medium heat. After it melts, continue to cook the butter, watching carefully. It will foam and subside, and then separate into golden butterfat and cloudy white milk solids. The milk solids will begin to brown. When they are lightly browned and the butter smells nutty and toasted, remove from the heat and set aside to cool.
In a mixer fitted with a whisk attachment (or using a hand mixer), whip the egg whites and sugar until stiff and glossy. Add the flour, vanilla, and brown butter and mix at low speed until blended. The batter will become thicker as the egg whites lose volume.
Using an offset spatula, an icing spatula, or the back of a spoon, spread the batter in circles to form thin disks on silicone baking mat-lined baking sheets. (You may only fit 6 or 8 per pan.) The batter should be so thin that you can see the pan through it.
Bake until light golden brown, 8 to 10 minutes. When the cookies come out of the oven, immediately lift them off the sheet pan with your fingers or a spatula and roll them up on a wooden spoon handle; then slide them off.
Repeat with the remaining batter. The batter will probably start to stiffen up as the butter cools; you can reheat it in the microwave, or set the bowl over hot water to keep the batter warm.