Melt 1 cup of butter in a small pot over medium heat, approximately 20 minutes until brown. Strain butter through mesh strainer and discard solids. Allow remaining butter to cool while you prepare the rest of the dough. Use butter to grease 24 3-inch Madeleine molds.*
Preheat oven to 375 degrees.
Combine eggs and vanilla in large mixing bowl. Beat with electric mixer for 3-4 minutes, until light and fluffy. Mix powdered sugar into egg and vanilla mixture slowly. When combined, beat with electric mixer for about 5 minutes, until smooth and thick. Combine flour and baking powder in small bowl. Mix well. Pour 1/2 half flour mixture into egg mixture. Stir. Repeat with 2nd half flour mixture. Stir well. Add cooled butter and stir until combined.
Use spoon to fill madeleine molds, filling each mold until it is 2/3 to 3/4 full. Put in oven, and bake for about 12 minutes, or until the edges are golden.
Remove from oven and molds immediately. Allow cookies to cool inverted on racks.
*Note: If you do not have a madeleine mold pan, you can use a mini muffin pan. You won't get the traditional madeleine shape, but the result will taste just as good.