Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.

In a medium bowl, combine the brown sugar, granulated sugar, and softened butter; beat until creamy.

Add in the pumpkin puree, egg, and vanilla extract. Mix well until fully combined.

In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.

Gradually add the dry ingredients to the wet mixture, stirring just until combined. Be careful not to over-mix!

Scoop tablespoon-sized amounts of dough onto the prepared baking sheet, spacing them about 2 inches apart.

Bake in the preheated oven for 10-12 minutes, or until the edges are lightly golden. Let cool on the baking sheet for a few minutes before transferring to a wire rack.

While cookies cool, prepare your frosting by browning the butter in a saucepan over medium heat until it turns golden brown and has a nutty aroma. Let it cool slightly.

In a mixing bowl, combine the brown butter with powdered sugar, adding milk as needed, to achieve a smooth consistency. Stir in vanilla.

Once the cookies have cooled completely, spread an even layer of brown butter frosting on top and enjoy!