Preheat the oven to 350 degrees F (175 degrees C) and lightly grease a 9-inch Bundt pan.

In a medium bowl, combine flour, cocoa powder, pumpkin pie spice, baking powder, and baking soda.

In a large bowl, beat together 3/4 cup butter, 2 cups sugar, applesauce, and eggs. Mix in 1/2 cup heavy cream and pumpkin until well combined.

Stir the dry flour mixture into the wet ingredients just until blended, then spread the batter evenly in the prepared pan.

Bake for 40 minutes in the preheated oven, or until a toothpick inserted into the center of the cake comes out clean. Let it cool in the pan on a wire rack before inverting onto a serving plate.

In a medium saucepan, combine brown sugar, 1/2 cup butter, and 1/3 cup heavy cream. Bring to a boil while stirring until smooth and the sugar is dissolved.

Whisk in the confectioners' sugar until smooth, then drizzle the frosting over the cake immediately.