Melt the butter first in the microwave so it will have time to cool.
Preheat your oven to 375℉. Spray your madeleine pan with non-stick cooking spray. Note: If you let the madeleines chill for 1 hour, wait until you are ready to bake.
In a small bowl, whisk together the flour, baking powder, salt and pumpkin spice. Set aside.
In the bowl to your stand mixer, beat the eggs and brown sugar together on medium-high speed until light and bubbly, about 4 minutes. You can alway use an electric mixer.
Add the pumpkin puree and vanilla and beat on medium-high for another minute.
Slow the mixer down to low and slowly add the flour mixture, a few tablespoons at a time, until just combined.
Add the melted butter and use a rubber spatula to fold it into the batter.
Optional - Chill the madeleine batter for an hour to enhance the flavor.
Scoop the pumpkin batter into the cavities of your greased madeleine pan filling each one about ¾ full.
Bake in your preheated oven for 10-12 minutes. The cakes will fill the shell shape, and the edges will appear be crispy. There should be a slight hump in the middle of each cake. Cool in the pan for 5 minutes before flipping out on a cooling rack to cool completely.