Heat the oven to 350°F. Grease a 9-by-13-inch metal pan and line the bottom with parchment.
Whisk flour, cinnamon, baking powder, baking soda, ginger, nutmeg and salt until evenly distributed.
Whisk pumpkin, oil, both sugars, eggs and 1 teaspoon vanilla until smooth and fully blended.
Fold dry ingredients into the pumpkin mixture just until the last flour streak disappears. Spread evenly in the prepared pan.
Bake for 28 to 36 minutes, until the center springs back lightly and a pick inserted away from a wet pumpkin pocket has a few moist crumbs. Cool completely.
Beat cream cheese and butter until smooth, then beat in confectioners' sugar and remaining vanilla. Add milk only if needed and spread over the cold cake.