Place butter into a small saucepan. Gently steep over low heat for 10 minutes to release the aromatic oils; do not boil. Remove from the heat, and set aside to cool for 30 minutes.

In a large bowl, whisk the eggs with the sugar until light colored and thick enough that it falls in ribbons. Depending on your mixer, this takes 5 to 8 minutes. Beat in the vanilla paste.

Fold flour into eggs, using a rubber spatula. Fold in the butter until just incorporated; don’t over-fold. Cover bowl with plastic wrap and chill in the refrigerator for at least an hour.

Preheat the oven to 375 degrees. Using a small pastry brush, grease two 12-form Madeleine molds with 2 tablespoons of melted clarified butter, spreading evenly. Refrigerate the molds until the butter is set, about 10 minutes.

Spoon rounded spoons of the batter into the center of the molds. Do not smooth batter, and fill each mold only about 2/3 full. Bake in the center of the oven for 10 – 12 minutes. The Madeleines should be gold in the center and lightly browned around the edges.

Remove from the oven and sharply rap the mold against a surface to loosen them. Turn out, and cool slightly on a wire rack. When Madeleines are barely warm, dust with powdered sugar.