In a heavy bottomed sauce pan, melt butter over medium heat. Once the butter has melted, continue to cook. Swirl pan occasionally to ensure even heating. Mixture will bubble and foam as it cooks. The bubbling will be vigorous at the beginning but it will subside. Continue to cook until butter smells nutty. The mild solids in the butter will turn rich brown in color. Remove from heat and pour into a heat-safe bowl. Be sure to scrape out as much of the browned milk solids bits.
Allow butter to chill in the fridge for 20-30 minutes until no longer warm to the touch. Butter will no solidify at this point. As long as butter is no longer hot, you may proceed to the next step.
Preheat oven to 350° F. Line 2- 12 cup muffin tins with cupcake liners. Set aside.
In the bowl of a stand mixer fitted with paddle attachment, beat together brown butter, sugar, and brown sugar until smooth.
Add eggs one at a time, mixing on low speed until thoroughly combined. Scrape down bowl as needed. Add butternut square puree and mix until incorporated.
In a large bowl, whisk together flour, baking soda, baking powder, cinnamon, nutmeg, and salt. Add to mixing bowl in two additions. Mix on low-medium speed until there are no longer any dry streaks of flour. Scrape down bowl as needed.
Distribute batter among prepared muffin tin. Bake for 25-30 minutes until toothpick inserted in center of cake comes out clean. Allow cupcakes to cool in tin for 5 minutes. Transfer cupcakes to wire rack to cool to room temperature.