Preheat your oven to 375°F (190°C) and lightly butter the madeleine mold to prevent sticking.
In a large mixing bowl, whisk together the eggs and sugar until the mixture becomes pale, thick, and slightly bubbly—about 3-4 minutes. This incorporates air for a light crumb.
Gently fold in the orange zest and ground cardamom, blending evenly into the egg mixture to distribute the flavors.
Slowly pour in the melted butter, folding carefully to combine without deflating the mixture, keeping it airy and smooth.
Sift the flour and salt together, then gently fold into the wet mixture in three additions. Use a spatula to avoid knocking out the air, resulting in a smooth, thick batter.
Transfer the batter into a piping bag or a plastic zip-top bag with the tip cut off. Pipe or spoon the batter into the prepared madeleine mold, filling each cavity about three-quarters full.
Bake the madeleines in the preheated oven for 9-11 minutes, or until the edges are golden brown and the shells have formed a characteristic hump.
Remove the madeleines from the oven and carefully transfer them to a wire rack. Let them cool for a few minutes until they firm up slightly before removing from the mold.
Once cooled slightly, gently tap each madeleine to loosen from the mold and serve warm or at room temperature to enjoy their tender, buttery crumb with a crisp exterior.
