Step 1: Preheat your oven to 350°F (175°C). Grease and flour two 9-inch round cake pans or a 9x13 inch baking pan. Line the bottoms with parchment paper if using round pans.
Step 2: In a large bowl, whisk together the flour, baking soda, cinnamon, nutmeg, ginger, cloves, and salt. Set aside.
Step 3: In a separate large bowl, combine the granulated sugar and brown sugar. Beat in the eggs one at a time until well combined. Stir in the vegetable oil, pumpkin puree, and vanilla extract until smooth.
Step 4: Gradually add the dry ingredients to the wet ingredients, mixing on low speed or by hand until just combined. Be careful not to overmix.
Step 5: Divide the batter evenly between the prepared cake pans (or pour into the 9x13 inch pan). Bake for 30-35 minutes for round cakes, or 40-45 minutes for a 9x13 inch cake, or until a wooden skewer inserted into the center comes out clean. Let cool in the pans for 10 minutes, then invert onto a wire rack to cool completely before frosting.
Step 6: Prepare the Brown Butter Frosting: In a light-colored saucepan, melt the 1 cup of unsalted butter over medium heat. Continue to cook, swirling occasionally, until the butter foams and the milk solids at the bottom turn golden brown and smell nutty. Remove from heat immediately and pour into a heatproof bowl to cool for 15-20 minutes until slightly solidified but still soft.
Step 7: Transfer the cooled brown butter to a large mixing bowl. Beat with an electric mixer until creamy. Gradually add the sifted powdered sugar, alternating with the milk or heavy cream, until smooth and fluffy. Beat in the vanilla extract and a pinch of salt. If frosting is too thick, add a bit more milk; if too thin, add more powdered sugar.
Step 8: Once the cake layers are completely cool, frost the top of one layer, place the second layer on top, and then frost the top and sides of the entire cake. Slice and serve!