In the bowl of a stand mixer fitted with the paddle attachment, beat the eggs and sugar together on medium-high speed until they are smooth, pale yellow, and creamy, about 8 minutes.
2 large eggs, ½ cup granulated sugar
Using a rubber spatula, gently fold in the flour, pumpkin puree, pumpkin pie spice, and baking powder.
1 cup all-purpose flour, ¼ cup pumpkin puree, 1 teaspoon pumpkin pie spice, ½ teaspoon baking powder
Add the melted and cooled butter (it should still be liquid, but not warm to the touch). Fold the butter into the batter gently.
½ cup unsalted butter
Transfer the batter to the refrigerator to chill for 30 minutes.
Towards the end of the chilling step, preheat oven to 350°F. Grease a madeleine pan with melted butter and nonstick spray.
Scoop 1 tablespoon of batter into each mold, making sure to fill each one about ¾ of the way full.
Bake in a preheated oven until the madeleines are light brown and a toothpick inserted into the center comes out clean, about 10-12 minutes.
Transfer the madeleines to a wire rack to cool.