Preheat oven to 180C/350F and grease a 12 hole muffin tin with butter.
Mix the sourdough starter, eggs, buttermilk, vanilla extract and oil in a bowl. Whisk together until well combined. It's ok if the oil sits on the top, it will combine once it is poured into the dry ingredients.
In a separate bowl mix together shredded carrot, all purpose flour, salt, baking powder, baking soda, spices and brown sugar.
Now pour the liquid ingredient mixture into the dry ingredients. Gently fold together until just mixed - don't over mix or the muffins will be tough.(see notes for options here)
Spoon mixture into greased muffin tin. This recipe will fill a 12 hole muffin (each hole around ¾ full).
Bake at 180C/350F for approximately 20 minutes or until muffins are golden brown.
Allow to cool in the tin for 10 minutes before removing and cooling further on a wire rack.
Add all frosting ingredients to the bowl of a stand mixer and beat for around 2 to 3 minutes or until thick and glossy. Hand mixer works well too.You can mix this frosting by hand, but you will need to really work the cream cheese and butter to get them to combine.
