Preheat the oven to 350°F (175°C). Grease and flour a 9×13-inch baking pan.
In a bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger.
In a separate large bowl, mix melted butter, granulated sugar, and brown sugar until well combined. Add eggs, one at a time, followed by vanilla extract, pumpkin puree, and milk.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 30-35 minutes or until a toothpick inserted into the center comes out clean. Allow the cake to cool completely.
For the browned butter frosting, melt butter in a saucepan over medium heat. Cook until the butter turns a golden brown color and gives off a nutty aroma. Remove from heat and let it cool for a few minutes.
In a mixing bowl, whisk together powdered sugar, vanilla extract, and browned butter. Add milk, a tablespoon at a time, until the desired consistency is reached.
Spread the browned butter frosting over the cooled cake.
