Preheat oven to 350°F (175°C) and line a 12-cup muffin tin with cupcake liners.
Whisk pumpkin purée, eggs, oil, both sugars, and vanilla until smooth. In another bowl, whisk flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves. Gently fold dry ingredients into wet until just combined. Fill liners about ⅔ full.
Bake 18–22 minutes until a toothpick comes out clean. Cool in the pan for 5 minutes, then transfer to a wire rack to cool completely.
Beat cream cheese and butter until fluffy. Gradually mix in powdered sugar, then add vanilla and a pinch of salt. Chill frosting for 15 minutes if too soft.
Frost cooled cupcakes and garnish with chopped nuts, a dusting of cinnamon or pumpkin spice, or a caramel drizzle.