To make the cheesecake layer, beat together the cream cheese, butter and sugar until smooth. Beat in the egg and vanilla just until combined. Set aside.
Grease and flour a 10 inch bundt pan. Preheat oven to 350 F. In a large bowl, beat together the sugar, oil and eggs until fluffy. In a separate bowl, stir together the flour, baking soda, pumpkin pie spice and salt. Add to the wet ingredients, alternating with the pumpkin. Stir until well combined.
Pour half the batter into the greased bundt pan. Top with the cheesecake mixture, being careful to keep it away from the edges of the pan. Top with the remaining pumpkin batter. Bake for 60-65 minutes or until a toothpick inserted comes out clean.
Let cool in the pan for 10 minutes before inverting onto a wire rack to cool completely.
To make the glaze, brown the butter in a saucepan over medium heat, stirring occasionally. Let cool. Mix in the powdered sugar, cream and as much milk as necessary to make a smooth glaze. Pour over the cooled cake. Store in the refrigerator.