Preheat the oven to 180°C. Line a standard 12-hole muffin tray (Yorkshire pudding tray) with paper cases.
Add the eggs and granulated sugar to a large bowl and whisk together using a hand-held electric whisk until combined.
Sift the gluten-free flour, baking soda, baking powder, cinnamon, vanilla sugar and salt into the egg and sugar mixture.
Stir to combine, then add the oil and grated carrots and gently fold the mixture until you have a smooth batter.
Divide the mixture evenly between the 12 muffin cases.
Bake for 18–25 minutes, or until a skewer inserted into the centre comes out clean.
Remove from the oven and allow the muffins to cool completely before adding the topping.
Add the cream cheese and butter to a large bowl and beat with a hand-held electric mixer until creamy and completely smooth.
Add the vanilla essence and salt and mix again until well combined.
Add the vanilla essence and salt and mix again until well combined.
Spoon the frosting into a piping bag fitted with a large open star nozzle.
Once the muffins are completely cool, pipe the frosting on top and finish with a sprinkle of chopped walnuts.