Melt the 1 1/2 sticks of butter in a small pot over medium heat until it's brown and gives off a deliciously nutty aroma, roughly 20 minutes.
Strain (using a paper towel over a mesh strainer) - you want to leave the solids behind.
Cool the browned butter to room temperature.
Beat the eggs and salt with a whisk attachment until it doubles or triples in size - approximately 3 minutes.
Continuing to mix on high speed, slowly add the sugar in a steady stream.
Add the vanilla and whip for 2 minutes or until mixture is thick and ribbony.
Sprinkle the flour on top of the egg batter, and gently fold in.
Now fold in the butter mixture. Only stirring enough to bring everything together.
Spoon the batter into the molds, bake the for 12 - 14 minutes (7-10 minutes for smaller cookies), or until the edges of the Madelaine's are golden brown.