Preheat and prep. Preheat the oven to 400°F. Grease a 10-inch bundt pan with baking spray or butter. In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, and salt.
Make the batter. In the bowl of a standing mixer with the paddle attachment, or with a hand mixer and beaters, cream the butter until pale and fluffy. Add the brown sugar and vanilla, mix together on medium speed until light in color. Add the eggs one at a time mixing until fully incorporated. Mix in the pumpkin puree, scraping down the sides and bottom of the bowl halfway. Add the dry ingredients on low-speed mixing until just combined and scraping down the sides of the bowl if needed.
Pour the batter. Place the cream cheese in a small bowl and microwave for about 30 seconds until very soft. (Note: this step is not necessary, but it makes the cream cheese swirl inside the cake smooth. You could just cut and place the cream cheese.). Pour half the batter inside the bundt pan. Sprinkle it with half the chopped chocolate, coat with the chocolate with the soft cream cheese, lightly smoothing it with a spatula. Top with the remaining chocolate pieces followed by the rest of the batter.
Bake. Bake for 35-40 minutes at 400°F or until a toothpick inserted in the center of the cake comes out clean or with just a few crumbs. Let the cake cool slightly, then place a piece of parchment paper or a plate over the bundt pan and gently flip over waiting for the cake to release from the pan. Let cool for about 15 minutes before adding the glaze.
Glaze the cake. In a small bowl, whisk together the confectioner's sugar and cocoa powder. Stir in the bourbon, then the heavy cream. Whisk until smooth and pour over the top of the cake.
Serve. Best if served slightly warm or at room temperature. Refrigerate in between servings.