Step 1: Place the butter in a medium saucepan over medium heat. Melt and continue cooking, stirring frequently, until the butter foams and brown bits appear at the bottom. Once it smells nutty and turns amber, remove from heat and let cool for 10 minutes.
Step 2: Preheat your oven to 350°F (175°C) and grease a 9x13 inch baking pan or line it with parchment paper.
Step 3: In a medium bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, ginger, and cloves until well combined.
Step 4: In a large mixing bowl, whisk the cooled brown butter with the granulated sugar and brown sugar until smooth.
Step 5: Add the eggs one at a time, whisking well after each addition. Stir in the pumpkin puree and vanilla extract.
Step 6: Gradually add the dry ingredients to the wet ingredients, folding gently with a spatula just until no streaks of flour remain. Do not overmix.
Step 7: Pour the batter into the prepared pan and spread evenly. Bake for 40-45 minutes, or until a tester comes out clean.
Step 8: Allow the pumpkin brown butter cake to cool completely in the pan on a wire rack before frosting or serving.