Preheat oven to 350 degrees. Line a cupcake pan with paper liners.
In a large bowl, add in the butter, oil, brown sugar, and granulated sugar. Beat with a hand mixer on medium until creamed. Scrape down the sides of the bowl.
Add in the vanilla extract. Next, add in the eggs one at a time and beat until each egg is completely incorporated. Scrape down the sides of the bowl after each egg.
Add in the pumpkin puree and beat until incorporated.
In a separate bowl, add in pumpkin pie spice, cinnamon, cloves, ginger, baking soda, baking powder, and flour. Whisk until combined.
Add in the dry ingredients in with the wet and beat on low just until the flour is worked into the batter.
Fill each cupcake liner about ¾ full of batter.
Place into the oven for 20 - 25 minutes or until done. Place a toothpick into the center of the cupcake and if it comes out clean or with a few moist crumbs, it's done!
Let cool in the pan for 10 minutes and let them finish cooling on a cooling rack.
Frosting
In a large bowl, add in the cream cheese and butter. Beat with a hand mixer on medium until creamy.
Add in the powdered sugar and milk. Beat until incorporated. Scrape down the sides of the bowl and beat in any unmixed bits.
Add the frosting into a piping bag with a straight tip and pipe swirls on the cooled cupcakes.