For the Brown Butter:

1. Chop 1/4 cup of the butter into even squares. Melt the butter in a small pot over medium-low heat.
2. Stirring the butter constantly with a whisk, continue to cook the butter until it starts foaming. Keep stirring until the butter turns a deep golden color, develops dark flecks, and releases a nutty aroma.
3. Immediately remove from the heat and pour the browned butter into a heatproof clean bowl through a fine-mesh strainer. Set the browned butter aside to cool.

For the Caramelized Pecans:

1. In a separate small pan, toast the pecans with about half a tablespoon of butter on low heat. As they begin to brown, add 1 tablespoon of brown sugar and stir constantly.
2. Remove them from the pan immediately after everything is mixed together and the brown sugar has completely melted. Set aside to harden and cool completely as you make the cake.

For the Cake:

1. Preheat the oven to 350°F. Melt the remaining 1 1/2 tablespoons of butter. Generously grease two madeleine pans with the melted butter.
2. In a small bowl, mix the flour, baking powder, and pumpkin spice together.
3. In a stand mixer fitted with a whisk attachment, beat the eggs together with a pinch of salt until they’re pale yellow, thick, and syrupy, approximately 10 minutes. While the mixer is still running on low speed, gradually add 1/3 cup brown sugar. When all the sugar has been incorporated, increase the speed to high. Continue beating until the mixture has gained even more volume, similar to softly whipped cream, about 5 minutes.
4. Remove the bowl from the mixer. Sift the flour mixture evenly on top of the whipped eggs, and gently fold it in with a spatula. Once the flour is completely mixed in, fold in the browned butter and pumpkin puree until completely incorporated. Fold in the caramelized pecans.
5. Place a heaping tablespoon of batter in each mold of the prepared pans. You will get 18 cookies, so not all the molds will be used in the second pan. Use the back of a spoon to spread the batter evenly in the molds. They should be almost filled, but with just a little room to expand.
6. Bake for 12-15 minutes, turning the pan once halfway through. They will be golden in color and will spring back when gently pressed down with your fingertip.
7. Remove them from the oven and let cool for 10 minutes before removing from the pan and inverting them onto a cooling rack to cool completely.
8. Dust with powdered sugar and serve immediately.