Preheat the oven to 175 degrees Celsius.
Prepare 8 muffin tins.
Combine milk and apple cider vinegar
Add apple sauce, coconut/olive oil, brown sugar, granulated sugar to a bowl and whisk until incorporated and smooth.
Combine oat flour, rice flour, baking soda, salt and cinnamon in a bowl.
Add the dry ingredients to the wet and whisk just until combined.
Grate the carrots and squeeze out excess water.
Fold into the batter using a spitula.
Divide the batter between muffin cups. Place in the middle of the oven and bake for 18-21 minutes, until a inserted toothpick comes out clean.
Remove from oven and let them cool completely.
Make the frosting by adding all ingredients in a bowl and whisk until light and fluffy. Place in the fridge for 20 minutes, or until ready to server.
Pip the frosting on top of each cupcake using a piping bag or just using a spoon.
Top with chopped walnuts (if using).