Preheat oven to 350 degrees. Line one regular size muffin tin with liners, set aside.
In the bowl of a stand mixer, beat together butter, oil, and sugars for 2-3 minutes or until light and creamy. Add pumpkin puree and vanilla, then with mixing speed on low, add one egg at a time, mixing until incorporated.
In a medium size mixing bowl, whisk together flour, baking powder, baking soda, salt, pumpkin pie spice, and cinnamon. With mixing speed on low, gradually add dry ingredients, mixing until just combined.
Remove bowl from stand and fill each liner ¾ of the way full. Place in oven and bake for 18-22 minutes or until the tops are lightly browned and spring back when touched. Remove from oven and allow to cool in pan for 3-4 minutes before transferring to a wire rack to cool completely (about 30 minutes).
While the cupcakes are baking prepare the salted caramel. Heat the granulated sugar in a small saucepan over medium heat, stirring constantly to prevent burning. The sugar will begin to form clumps and will eventually melt and turn brown. Once the sugar has melted completely, add the butter. The mixture will bubble, continue stirring for 1-2 minutes or until the butter has completely melted, then add the cream. Boil for 1 minute then remove from the heat and add the salt. Allow the caramel to cool while the cupcakes finish baking/cooling.
To prepare the cream cheese frosting, using the paddle attachment cream the butter, vanilla, salted caramel and salt for 2 minutes or until light and fluffy. With mixing speed on low gradually add the powdered sugar, beat until it forms a thick paste. Add cream cheese and mix on the lowest speed possible for 1½ minutes, then STOP! Place in refrigerator to chill for at least 15 minutes or until cupcakes have completely cooled. Allow to stand at room temperature for 5 minutes before placing in a bag to pipe on cupcakes. Garnish cupcakes with a drizzle of slightly warm salted caramel (warm enough for it to be fluid but not to melt the frosting) and a sprinkle of sea salt.