Preheat oven to 350℉. Line a 9-by-13-inch metal baking pan with parchment paper, allowing it to overhang the edges. Coat with cooking spray.
Whisk pastry flour, all-purpose flour, pumpkin pie spice, baking powder, baking soda and salt in a large bowl. Add pumpkin, water, granulated sugar, brown sugar, oil and ½ teaspoon vanilla; whisk until smooth.
Scrape the batter into the prepared pan and spread into an even layer. Bake until a toothpick inserted in the center comes out clean, 25 to 30 minutes. Let cool completely in the pan on a wire rack for 1 hour before frosting.
Place cream cheese and butter in a medium bowl; beat with an electric mixer on high speed until smooth. Slowly add ½ cup confectioners' sugar; beat on medium speed to combine.
Add another ½ cup confectioners' sugar and the remaining 1 teaspoon vanilla; beat to combine. Add the remaining 1 cup confectioners' sugar; beat on high speed until smooth and fluffy.
Spread the frosting in an even layer over the cooled cake.
