Make the pumpkin puree: Position a rack in the center of the oven and heat the oven to 350°F Brush a 9x13-inch baking dish with the oil. Put the pumpkin halves in the dish cut side down and bake until tender when pierced with a fork, about 45 minutes. Let cool. Peel the pumpkin and puree the flesh in a food processor until smooth. You'll need 1-1/2 cups of the puree for the cake. Refrigerate or freeze any remaining puree for another use. Make the cake: Position a rack in the center of the oven and heat the oven to 350°F. 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 puree 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. Make the topping: Melt the butter in a heavy-duty 12-inch nonstick skillet over medium heat. Add the pecans and pepitas and cook until the pecans brown slightly and the pepitas begin to pop, about 2 minutes. Sprinkle in the brown sugar and salt and stir until the sugar melts and the nuts are glazed, about 2 minutes. Stir in the ginger. Remove from the heat and let the mixture cool in the skillet. Make the frosting: 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 the solids settle at the bottom of the bowl, about 5 minutes. Carefully transfer the bowl to the freezer and chill until just firm, about 18 minutes. Using a spoon, carefully scrape the butter from bowl, leaving the browned solids at the bottom; discard the solids. 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. Assemble the cake: Put one cake layer on a cake plate. Spread 1/2 cup of the frosting on the layer. Sprinkle 1/2 cup of the nut mixture over the frosting and top with the second layer. Frost the top and sides of the cake with the remaining frosting. Arrange the remaining topping in a ring 1-1/2 inches in from the edge of the cake and serve. Make Ahead Tips: You can make the puree up to 2 days ahead. The assembled, frosted cake can be covered with a cake dome and refrigerated for up to 2 days. Serve at room temperature.