Make the cake: Position a rack in the center of the oven and heat the oven to 325F. Grease a 9 x 13-inch baking dish.
Add the cranberry sauce to a large bowl and use a whisk to break it up into small chunks. Add the sugar, vegetable oil, and vanilla and whisk to combine. Whisk in the milk and eggs.
In a medium bowl, whisk together the cake flour, baking powder, baking soda, salt, and cinnamon.
Add the dry ingredients to the wet and whisk until smooth. Pour the batter into the prepared dish.
Bake, rotating the pan front to back halfway through, or until a toothpick inserted into the center of the cake comes out clean or with a few crumbs clinging to it, 35-40 minutes. Transfer the dish to a rack to cool completely before frosting.
Make the cream cheese frosting: In a stand mixer fitted with the paddle (or in a large bowl if using an electric hand mixer), beat the butter on medium-high speed until it’s light and fluffy, about 5 minutes. Stop to scrape down the sides of the bowl to ensure all the butter gets creamed.
On medium speed, beat in the cream cheese and vanilla until just combined, about 3 minutes more. Turn the mixer to low speed and add the powdered sugar, about 1 cup at a time, and mix until incorporated.
Scoop the frosting onto the cooled cake. Spread and swirl to cover. Leftover cake can be stored in an airtight container at room temperature for up to a day or in the fridge for up to three days.