Cook the full can of pumpkin puree (about 1 3/4 cups) in a saucepan over medium heat, stirring frequently, for 8 to 10 minutes until thickened, slightly darker, and reduced to about 1 1/4 cups. Cool completely.
Preheat oven to 350 degrees F. Grease and flour three cake pans and line the bottoms with parchment rounds.
Whisk the cake flour, pumpkin pie spice, baking soda, baking powder, and salt in a medium bowl.
In a large bowl, whisk the cooled reduced pumpkin puree, brown sugar, granulated sugar, oil, eggs, and vanilla until smooth.
Fold the dry ingredients into the wet mixture just until no flour streaks remain. Mix in the milk, one tablespoon at a time, if the batter needs loosening.
Divide the batter among the three pans and bake for 26 to 30 minutes, until a toothpick comes out clean.
Cool in the pans for 10 minutes, then transfer to a wire rack to cool completely before frosting.
Beat the cream cheese and butter until smooth. Add powdered sugar gradually, then mix in vanilla, cinnamon, and salt until fluffy.
Frost between each layer, apply a thin crumb coat, refrigerate for 20 minutes, then apply the final coat of frosting.
Slice and serve, or refrigerate until ready to serve.