Preheat the oven to 350°F (177°C). Grease an 8-inch square metal cake pan and line the bottom with parchment paper.
Whisk the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, cloves, and salt in a medium bowl.
In a large bowl, whisk the pumpkin puree, vegetable oil, granulated sugar, and brown sugar until combined.
Add the eggs one at a time, whisking after each addition. Mix in the milk and vanilla flavouring.
Fold the dry ingredients into the wet ingredients until just combined. Avoid overmixing.
Spread the batter evenly in the prepared pan.
Bake for 30 to 38 minutes, checking from 28 minutes onward. A toothpick inserted into the centre should come out clean or with a few moist crumbs.
Cool the cake in the pan for 15 minutes, then transfer it to a wire rack and cool completely.
Beat the cream cheese and butter for 1 to 2 minutes until smooth.
Add the powdered sugar gradually, then mix in the vanilla flavouring. Add milk a teaspoon at a time only if needed.
Spread the frosting over the completely cooled cake.
Chill briefly for neater slices, cut into 12 portions, and refrigerate leftovers promptly.