Place the eggs and sugar in the bowl of a stand mixer fitted with the whisk attachment. Mix on high until the batter is thick enough to hold a figure-8 before melting back into the batter (see notes). This will take 5 to 8 minutes.
Add the vanilla and mix to combine.
Sift the flour, cocoa, baking powder, and salt straight into the bowl. Use the whisk attachment to fold the ingredients together (by hand).
Add in about half of the butter and stir to combine. Add in the remaining butter and fold together until combined.
Cover the bowl and place in the refrigerator for at least 30 minutes (preferably 2 hours) or up to one day.
About 30 minutes before baking, brush the inside of a madeleine pan with melted butter. Light dust it with flour then tap out the excess. Place the prepared pan in the freezer for at least 15 minutes.
Meanwhile, preheat the oven to 350°F.
When ready, pipe the batter into the deepest part of the cavities of the prepared pan - about 2 tablespoons of batter each. Do not flatten the batter.
Bake for 10 to 12 minutes, or until the madeleines spring back to the touch and a toothpick inserted into the center comes out clean-ish. Allow to cool for about 10 minutes then tip the baked madeleines out onto a rimmed baking sheet.
For the chocolate glaze, melted the chocolate with the oil of your choice (in the microwave or over a double-boiler).
Clean the Madeleine pan, the spoon about a tablespoon of melted chocolate into each cavity. Use the back of the spoon to spread it out.
Pop the baked madeleines back into the pan and press into the chocolate to coat.
Freeze the chocolate madeleines for 15 to 20 minutes before carefully popping them out of the pan. Press the edge of a small knife or offset spatula into one side of the Madeleine and they should pop out.
Serve madeleines the same day as they are made. Store leftovers in an air-tight container for up to a week.