Start by melting the butter and then set it aside to slowly come to room temperature.
In a large bowl, whisk the eggs, egg yolks, and sugar together.
Add the vegetable oil and lemon zest and whisk.
Add the flour, baking powder, grated Parmesan, salt, and pepper. Whisk.
Add the room-temperature melted butter. Whisk.
Whisk in the herbs very gently, just until they're incroporated into the batter.
Butter your madeleine tray with some softened butter. Fill each madeleine 3/4 of the way up with the batter. Put the tray in the fridge.
Preheat the oven to 210C.
Once the oven is hot, take the pan out of the fridge and place it directly in the oven.
Bake for 6 minutes, then turn down the temperature to 180C. Keep the oven door open for 10-15 seconds to help the internal temperature come down.
Bake for another 5-7 minutes, or until a toothpick inserted in the center comes out clean.
Immediately remove the madeleines from the tray onto a cutting board or flat surface, to cool down for a couple of minutes. Then, transfer to a wire rack to cool completely.
While the madeleines are cooling, make the horsradish cream by combining the cream with the horseradish, seasoning with salt and pepper to taste and mixing well.