Prep. Preheat your oven to 350°F (175°C) and place the rack in the middle. Then, coat a 10x15-inch jelly roll pan with nonstick cooking spray.
Make the batter. In a large mixing bowl, use an electric hand mixer on medium speed to beat together the canned pumpkin, sugar, and oil until well combined. Add the eggs one at a time, beating until each egg is fully incorporated and scraping down the bowl as necessary. In a separate bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, cloves, and allspice. Gradually add the dry ingredients to the pumpkin mixture, mixing on low speed until just combined.
Add starter. Mix in the sourdough starter by hand, making sure it's completely mixed into the batter. Scrape down the sides and bottom of the bowl so everything is well combined.
Bake and cool. Pour the batter into the prepared pan, spreading it out evenly. Bake in the preheated oven for 25-30 minutes or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan on a wire rack. After 15 minutes, cover the cake with a clean tea towel to prevent it from drying out.
Make the frosting. Once the cake is completely cooled, prepare the frosting. Blend the melted butter, cream cheese, and vanilla with a mixer until fluffy. Gradually add the powdered sugar, beating until smooth. If the frosting is too thick, add a splash of milk to reach the desired consistency.
Serve. Spread the frosting over the cooled cake and serve directly from the pan. Refrigerate any leftovers.