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Preheat the oven to 350°F (176°C). Spray a 12-cup bundt pan with non-stick cooking spray; set aside.
To make the Cream Cheese Filling: In a stand mixer, beat the cream cheese with the granulated sugar until smooth and light in texture. Add the egg, cinnamon and vanilla; beat in until smooth. Transfer the filling to a separate bowl and place in the fridge while preparing the cake batter.
To make the Pumpkin Bundt Cake: In a large bowl, whisk together the flour, baking powder, baking soda, salt and spices. In the bowl of a stand mixer, cream together the butter and sugars. Beat in the eggs one at a time. Add the vegetable oil, pumpkin, yogurt and vanilla; beat until the mixture is smooth. Mix in the dry ingredients on the lowest speed just until evenly incorporated.
Assemble by pouring half the pumpkin batter into the prepared bundt pan, making sure the batter covers the bottom of the pan and fills every crevice. Smooth the top with the back of a spatula. Add the cream cheese filling to cover the first layer of batter, again reaching every crevice and smoothing on top. Repeat with the remaining pumpkin batter, covering the cream cheese completely. Rap the pan firmly against the counter to ensure that the batter settles into the pan without any empty pockets.
Bake the cake in the preheated oven for 55-60 minutes or until a toothpick inserted into the middle area of the cake comes out clean. Let the bundt cool in the pan for 15 minutes, and then transfer it to a wire rack to cool fully.
Make the icing by microwaving the butter and milk in a heat-safe bowl for 40 seconds, or until the butter has melted. Add the powdered sugar and stir in until it has melted and the mixture is smooth. If thicker icing is desired, add more powdered sugar to achieve the preferred consistency. Milk can also be added to achieve a thinner icing. The icing should be somewhat thick but still runny enough to flow easily over the bundt cake.
Pour the icing carefully over the top of the cooled cake, focusing on the peak and letting it glaze down the sides. Work quickly because the glaze will set fast.