Preheat the oven to 350°F (177°C). Thoroughly grease a 10-inch Bundt pan, taking care to coat all ridges and crevices so the finished cake releases cleanly, including when using a nonstick pan.
In a large mixing bowl, whisk together the all-purpose flour, baking soda, fine sea salt, cinnamon, ginger, cloves, nutmeg, and allspice until the spices and leavening are evenly distributed. Set the bowl aside.
In a separate large bowl, whisk together the pumpkin purée, packed brown sugar, granulated sugar, vegetable oil, four eggs, and vanilla until the mixture is smooth and uniformly combined.
Finish the pumpkin batter. Pour the pumpkin mixture into the bowl containing the dry ingredients. Whisk until a thick, cohesive batter forms and no pockets of dry flour remain.
Place the softened cream cheese in the bowl of a stand mixer fitted with a paddle or whisk attachment, or use a handheld electric mixer. Beat on high speed for approximately 1 minute, until completely smooth and free of lumps. Scrape down the sides and bottom of the bowl.
Add the room-temperature egg, granulated sugar, and vanilla to the cream cheese. Beat on medium-high speed for about 1 minute, or until the filling is creamy and thoroughly blended.
Spoon approximately half of the pumpkin batter into the prepared Bundt pan and spread it into an even layer. Distribute the cream cheese mixture evenly over the pumpkin batter.
Cover the cream cheese layer with the rest of the pumpkin batter, gently smoothing the surface so the filling is enclosed.
Run a knife carefully through the layered mixtures with a gentle swirling motion to form a marbled pattern between the pumpkin batter and cream cheese filling.
Transfer the pan to the preheated oven and bake for 55–65 minutes. The cake is ready when a toothpick inserted into the cake portion comes out clean or with only a few lightly moist crumbs attached.
Place the Bundt pan on a wire cooling rack and allow the cake to cool inside the pan for 2 hours. After this resting period, carefully invert the cake onto a cooling rack, serving plate, or cake stand and allow it to cool completely.
Once fully cooled, drizzle the cake generously with salted caramel sauce. Sprinkle the surface with flaky sea salt, if using, before slicing and serving.