Prepare Brown Butter: In a medium saucepan, melt the butter over low-medium heat. Increase to high until boiling, then reduce to medium until it browns with a nutty aroma. Remove from heat and transfer to a heatproof container to cool.
Mix Dry Ingredients: In a medium-size mixing bowl, sift together the flour and baking powder. Whisk through the salt. Set aside.
Whisk Eggs and Sugar: In a separate large mixing bowl, whisk the eggs, sugar and vanilla for 2 minutes until pale yellow.
Combine Ingredients: Gently fold half the flour mixture into the eggs until just combined. Fold in the remaining flour.
Mix in Butter: Pour the cooled brown butter into the batter and gently fold until well combined, until the batter is thick, do not over-mix.
Chill Batter: Cover and refrigerate the batter for 30 minutes.
Preheat Oven and Prepare Pan: Preheat oven to 356ºF (180ºC). Brush madeleine pan wells with melted butter, dust with flour, and tap out excess.
Transer Batter: Spoon 1 tablespoon of batter into the middle of each well. Leaving the batter untouched, it will spread out into the wells during baking.
Bake: Bake for 10-12 minutes until golden and springy to the touch. Cool in the pan for 3 minutes, then transfer to a wire rack to cool completely.
Serve: Dust with powdered sugar and enjoy!