Preheat oven to 350 degrees. Grease and flour a 9x13" pan and set aside.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and pumpkin pie spice. Set aside.
In a large mixing bowl (or in the bowl of a stand mixer if you'd like to use one), whisk together the brown sugar, white sugar, oil, applesauce, and vanilla. Blend in eggs and mix until smooth, then add pumpkin and mix again. When the mixture is combined completely, slowly add in the flour mixture, beating until just combined. Pour into the prepared pan and even it out with a spatula. Bake for 30-35 minutes or until a pick inserted into the center comes out clean (times can vary widely depending on individual ovens).
When done, remove the cake from the oven and allow to cool completely.
While the cake is baking, prepare the frosting. Whip together the cream cheese, butter, and extract until light and fluffy. Add in the powdered sugar and whip until smooth and fully incorporated. when the cake is completely cool, spread with the frosting. Refrigerate any leftovers.