Butter and flour two madeleine plaques, or if you're using silicone pans, no need to prepare them.
Place plaques on a baking sheet.
Place the butter in a one quart microwaveable bowl. Cover with plastic wrap and heat in 20 second increments until melted. Set aside.
Ready your mixer with the whisk attachment. In the bowl of a stand mixer or large mixing bowl add the eggs, egg yolks and one tablespoon of the sugar. Beat until just combined. Add the remaining sugar and whisk on medium speed until mixture is airy, pale and tripled in volume. The eggs are properly whipped when the mixture falls back into the bowl in ribbos and rests on the surface for 10 seconds. If the ribbon sinks immediately, continue to whip the egg mixture until achieving the proper consistency.
Sprinkle about 1/3 of the flour into the egg mixture and gently fold it in with a rubber spatula, stopping as soon as the flour is incorporated. Add the rest of the flour in two more additions.
Gently spoon one cup of the batter into the bowl with the butter. Gently fold the butter into the batter, then add the butter mixture back into the larger bowl of batter and fold it very gingerly. This is the most fragile point for the batter, so don't over mix.
Drop rounded teaspoons of the batter into the madeleine molds to fill about 75-80% of the way, but don't overfill - they will puff as they bake.
Bake for 10 minutes until very lightly golden, and cake springs back when you touch it.
Cool the madeleines in the pan for 3-5 minutes until carefully unmolding them from the pan. Serve.
Note: Madeleines are best eaten the day that you make them or the next day, dipped in a cup of tea or coffee. They can also be frozen and thawed. To freeze, wrap the little cakes so they don't touch, in parchment paper. Place into a zip-top freezer bag. Freeze for up to a month.