Pumpkin Cake with Caramel-Cream Cheese Frosting

In a medium saucepan, combine the sugar, water, vanilla bean and seeds. Cook over high heat, stirring, until the sugar is dissolved. Using a wet pastry brush, wash down any crystals from the side of the pan. Cook over moderate heat without stirring until a medium-dark amber caramel forms, about 9 minutes. Remove from the heat and immediately stir in the butter and heavy cream. (Don't worry if the butter separates.) Discard the vanilla bean. Transfer the caramel to the large bowl of a standing mixer fitted with a whisk and beat at low speed until the caramel cools slightly and comes together, about 5 minutes. With the machine on, beat in the cream cheese, 1 cube at a time and beating well between additions, until silky. Transfer the frosting to a bowl and refrigerate until very firm, at least 6 hours. Preheat the oven to 350 °. Butter and flour two 8-inch round cake pans. In a medium bowl, whisk the flour with the baking powder, salt, cinnamon, ginger, baking soda, nutmeg and cloves. In a bowl, using an electric mixer, beat the brown sugar and eggs at medium-high speed until fluffy, 3 minutes. Beat in the oil, then beat in the pumpkin puree. Alternately add the dry ingredients and the milk in 3 batches, beating well between additions. Pour the batter into the prepared pans and smooth the tops. Bake for 40 to 45 minutes, until a toothpick inserted in the center of the cake comes out clean. Let the cakes cool on a rack for 20 minutes. Run a knife around the edges to loosen the cakes, then invert them onto a wire rack to cool completely. Place one layer on a plate and spread with 1 cup of the caramel–cream cheese frosting. Top with the second layer and frost the top and side. Refrigerate the cake for 2 hours before serving.