In a small saucepan, melt the butter over medium heat. Cook, stirring frequently, until the butter is golden brown and has a nutty aroma (about 5-7 minutes). Remove from heat and allow to cool slightly.
In a large mixing bowl, combine the brown butter, brown sugar, and granulated sugar. Whisk until smooth.
Add the pumpkin puree, maple syrup, egg, and vanilla extract to the butter mixture. Whisk until fully combined.
In a separate bowl, sift together the flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
Gradually add the dry ingredients to the wet mixture, stirring with a rubber spatula until just combined. Do not overmix.
Preheat your oven to 350°F (175°C). Line a cookie sheet with parchment paper.
Scoop tablespoons of dough onto the prepared cookie sheet, spacing them about 2 inches apart.
Bake in the preheated oven for 10-12 minutes, or until the edges start to turn golden brown. The centers will look slightly underbaked but will continue to cook as they cool.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a cooling rack.