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Place the butter in a medium 2-cup (16 fl oz/450 ml) capacity soup bowl and microwave for 15 seconds, or until melted.
To the bowl, add the flour, brown sugar, baking powder and cinnamon and stir with a fork.
Next whisk in the milk, grated carrot and vanilla and stir until combined and smooth.
Microwave on high for 2 minutes, then 15-20 seconds at a time, checking for doneness in between, until cooked through (the top will spring back if pressed when it's done). Let cool for 15 minutes and make the frosting.
Using an electric hand mixer, cream the cream cheese and butter together on high speed for 2 minutes, until light and fluffy.
Gradually add the powdered sugar a little at a time until combined. Lastly add in the vanilla extract.
Top the cake with the frosting and enjoy!!! Store leftovers covered at room temperature for up to 1 day. Any leftover cream cheese frosting will keep in the fridge for up to 7 days.