Melt 6 tablespoons of butter in a small pot over medium heat, making sure to stir constantly to avoid burning the butter.
When the butter is fully melted, add 1/2 teaspoon of orange zest and stir once or twice into the melted butter. Cool to room temperature.
Put the egg in the bowl of a freestanding electric mixer with a whisk attachment (or, use a handheld whisk. Whip on high speed until thick, approximately 3 minutes. The egg should double or triple in volume during this time.
Continuing to mix on high speed, slowly add the sugar in a steady stream. Continue mixing for 2 minutes, or until the mixture is thick. Do not rush through this step — it is important!
After the mixture is thick, turn off the mixer and use a spatula to gently mix in 1/4 teaspoon of vanilla extract.
Sprinkle 3 tablespoons of flour on top of the egg batter, and gently fold in. Stir only enough to integrate the flour, and no more.
Once the flour has been integrated into the egg batter, gently fold in the butter-orange zest mixture. Again, stir only enough to bring everything together and no more.
Cover the mixing bowl with saran wrap, and refrigerate for at least 30 minutes or more.
Once your batter has been chilled for at least 30 minutes, preheat your oven to 365 (F) — this is an increase from the original sea-level temperature of 350 (F). You will need to lower the oven temperature at sea-level.
Generously grease your madeleine pan. I used coconut spray oil, but butter is a perfectly viable alternative. Spoon the batter into the madeleine flutes, filling each 2/3-3/4 full. I filled each madeleine flute with a scant 1 tablespoon of batter, using a cookie scoop. DO NOT OVERFILL THE MADELEINE FLUTES, or butter will leach out of the pan and onto the oven floor, causing lots of scary smoke. I promise. Bake the madeleines for 12-14 minutes, or until the edges of the madeleines are golden brown. Remove from the oven and let rest for a minute or two before unmolding.