Preheat your oven to 375 degrees F and situate an oven rack in the middle of the oven. Grease a 10 - 12 inch springform pan with butter or a non-stick spray. Set aside.
To make the cake, in the bowl of your stand mixer with paddle attachment fixed, cream together the egg, egg yolk, butter, granulated sugar, and light brown sugar until combined. Add the buttermilk, pumpkin puree and vanilla extract and beat again until combined.
In a medium bowl, whisk together the flour, baking powder, pumpkin pie spice, ground cinnamon, and salt. With the mixer set on low-speed, slowly add the dry ingredients to the wet until all of the ingredients are combined. Do not over-mix. Scrape down the sides of the bowl as needed.
Pour the batter into the prepared pan. Set aside.
In a small bowl, whisk together the mascarpone cheese, granulated sugar and cinnamon until thoroughly combined. Spread the cheese mixture over the cake batter gently, so that it doesn't mix with the batter.
Prepare your crumble topping by combining the flour, brown sugar, cinnamon, nutmeg and allspice in a large bowl. Whisk to combine. Add the cold and cubed butter and crumble the ingredients together with either a pastry cutter, a fork or your hands until coarse crumbs form. Act quickly, so that the butter does not melt.
Sprinkle the crumble on top of the cheese mixture.
Bake the cake at 375 degrees F for 45-50 minutes or until a toothpick comes out clean. If you find that the cake is browning too much through the baking process, you may cover it with foil.
Remove from oven and let cool for at least an hour before serving. When cool, combine the confectioner's sugar and milk in a small bowl and whisk until smooth and creamy. Drizzle on cooled cake.
The cake will store at room temperature for 3 days. Enjoy!