Preheat your oven to 350°F (175°C). Grease and flour the 9x13-inch baking pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, spices, and salt. Set aside.
In a large mixing bowl, combine the pumpkin puree, granulated sugar, brown sugar, vegetable oil, eggs, and vanilla extract. Mix until smooth.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Pour the batter into the prepared baking pan and spread evenly. Bake for 30 minutes or until a toothpick inserted in the center comes out clean.
While the cake is baking, prepare the brown butter frosting. In a small saucepan, melt ½ cup of butter over medium heat. Continue to cook, stirring frequently, until the butter foams and turns a golden brown, about 5-7 minutes. Remove from heat and let cool slightly.
In a mixing bowl, whisk together the powdered sugar and vanilla extract. Gradually add the browned butter and mix until smooth. Add milk as needed to reach desired consistency.
Once the cake has cooled completely in the pan, frost with the brown butter frosting using a rubber spatula. Cut into squares and serve.