Butter and flour two 9-inch round cake pans with removable bottoms (or butter two 9-inch round cake pans, line the bottoms with parchment, butter the parchment, and flour the pans).
Melt the butter in a heavy-duty 1-quart saucepan over medium heat. Cook, swirling the pan occasionally until the butter turns a nutty golden-brown, about 4 minutes. Pour into a small bowl and let stand until cool but not set, about 15 minutes.
In a medium bowl, whisk the flour, baking soda, cinnamon, ginger, salt, and cloves. In a large bowl, whisk 1-1/2 cups of the pumpkin purée with the granulated sugar, brown sugar, eggs, and buttermilk until very well blended. With a rubber spatula, stir in the flour mixture until just combined. Gently whisk in the brown butter until completely incorporated. Divide the batter evenly between the prepared pans.
Bake the cakes until a tester inserted in the center comes out clean, about 28 minutes. Let the cakes cool in the pans for 10 minutes. Turn the cakes out onto racks, remove the pan bottoms or parchment, and cool completely.
Using an electric mixer, beat the butter, cream cheese, and brown sugar on medium-high speed until light in color and the brown sugar has dissolved, 2 minutes. Gradually beat in the confectioners’ sugar and continue beating until fluffy, 1 to 2 minutes.