To make the pumpkin puree - preheat the oven to 200°C. Slice the pumpkin in half, scoop out the seeds and the stringy flesh. Season lightly with salt and place the halves cut-side-down on a baking sheet lined with parchment paper. Roast until the flesh is soft, coming away from the skin and can be easily be pierced with a knife in the thickest part of the pumpkin, about 45 to 60 minutes. Cool until you can safely handle the halves, scoop the flesh into a food processor, working in batches if necessary. Process until smooth, about 3-5 minutes.
Turn down the oven heat to 180°C. Line a 12-cup muffin pan with cupcake liners.
Whisk flour, baking powder, baking soda, salt, ground cinnamon, allspice, ginger, nutmeg and cloves in a large bowl. Set aside.
Whisk oil, eggs, brown sugar, pumpkin and vanilla extract until combined. Pour the wet ingredients over the dry ingredients and whisk until combined.
Fill the cupcake liners 2/3 full with a spoon or a ice-cream scoop. Bake for 20-25 minutes or until a toothpick inserted in the center comes out clean. Let them cool completely before frosting.
To make the frosting, beat on high speed the cream cheese and butter until creamy. Add powdered sugar, vanilla extract and salt and mix on low speed for 30 seconds, then on high speed for 2 minutes until the sugar completely dissolves. You can add powdered sugar to thicken the frosting if it's necessary.
Frost cooled cupcakes with a piping bag. Store in the refrigerator.