Brown the butter. Add the butter to a light-colored saucepan over medium heat. Cook, stirring frequently, until it foams and the milk solids turn deep golden brown and smell nutty. Immediately pour the brown butter, including the browned milk solids, into a heat-safe bowl and cool for 15–20 minutes.
Prepare the pan. Preheat the oven to 350°F. Thoroughly grease a 10–12 cup Bundt pan, making sure to coat every ridge and the center tube. Dust lightly with flour and tap out any excess.
Mix the dry ingredients. In a medium bowl, whisk together the flour, pumpkin pie spice, cinnamon, baking powder, baking soda, and salt. Set aside.
Mix the wet ingredients. In a large bowl, whisk together the cooled brown butter, oil, and brown sugar. Add the eggs one at a time, whisking after each addition. Whisk in the pumpkin puree, sour cream, and vanilla until smooth.
Make the batter. Add the dry ingredients to the wet ingredients and gently fold with a rubber spatula just until no streaks of flour remain. Do not overmix.
Fill the pan. Spoon the batter evenly into the prepared Bundt pan and smooth the top. Gently tap the pan against the counter a few times to release any large air pockets.
Bake. Bake for 50–60 minutes, or until a long toothpick or skewer inserted into the thickest part of the cake comes out clean or with a few moist crumbs.
Cool. Let the cake cool in the pan for 15 minutes, then carefully invert it onto a wire rack. Allow the cake to cool completely before glazing.
Make the glaze. Beat the softened cream cheese and butter until smooth. Add the powdered sugar, maple syrup, vanilla, and salt and beat until creamy. If needed, add milk or heavy cream 1 teaspoon at a time until the glaze is thick but slowly pourable.
Glaze and serve. Spoon the maple cream cheese glaze generously over the completely cooled cake, allowing it to slowly drip down the sides. Let the glaze set for 15–20 minutes before slicing and serving.
