MAKE THE CRUST: Preheat the oven to 325°. Lightly butter a 9-inch springform pan. In a medium saucepan, melt the butter with the chopped chocolate over low heat, stirring constantly. Remove the melted chocolate mixture from the heat and let it cool slightly, then stir in the sugar and the lightly beaten egg until blended.
In a small bowl, whisk the flour with the baking powder and salt. Add the dry ingredients to the melted chocolate mixture. Stir the chopped walnuts into the brownie batter. Spread the batter in the prepared springform pan and smooth the surface with a spatula. Bake for 10 minutes, or until risen and dry to the touch. Let cool completely. Wrap the outside of the pan in a large sheet of foil. Leave the oven on.
MAKE THE CHEESECAKE: In a medium glass bowl, melt the chopped chocolate in a microwave oven on high for about 1 minute, stirring halfway through. In a large bowl, using an electric mixer, beat the cream cheese until smooth. Add the sugar and beat until light and fluffy. Add the eggs, 1 at a time, beating well between additions. Beat in the cornstarch and vanilla. Add 1 cup of the cheesecake batter to the melted chocolate and stir until well blended. Beat the pumpkin puree and the cinnamon, nutmeg and cloves into the remaining cheesecake batter.
Pour three-fourths of the pumpkin cheesecake batter over the brownie crust. Pour the chocolate batter on top of the pumpkin cheesecake batter. If the chocolate batter is too thick to pour easily, heat it in a microwave oven on high for 10 seconds. Spoon the remaining pumpkin cheesecake batter on top of the chocolate batter. Use a table knife to make a few decorative swirls: Do not overswirl.
Set the springform pan in a medium roasting pan. Add enough hot water to the roasting pan to reach halfway up the side of the springform pan. Bake the pumpkin cheesecake in the hot water bath in the center of the oven for 1 1/2 hours, or until it is firm around the edges but slightly jiggly in the center. Turn the oven off, prop the door several inches open and let the cheesecake stand in the water bath in the warm oven for 1 hour, or until it is completely set.
Remove the marbled pumpkin cheesecake from the water bath and refrigerate it until thoroughly chilled, at least 4 hours or overnight. Remove the foil and springform pan ring from the cheesecake. Carefully transfer the cheesecake to a platter, cut into wedges and serve.
Make Ahead: The baked cheesecake can be refrigerated in the springform pan for up to 4 days