Preheat the oven to 180°C conventional / 160°C fan / Gas Mark 4. Line a 12-hole muffin tin with suitable muffin cases.
Add the gluten-free self-raising flour, light brown sugar, cinnamon, ginger, nutmeg, bicarbonate of soda and salt to a large mixing bowl. Stir until evenly combined.
Add the finely grated carrot and stir until it is evenly distributed and lightly coated in the flour mixture.
In a separate bowl or jug, whisk together the apple purée, vegetable oil, orange juice and vanilla extract.
Pour the wet ingredients into the dry ingredients and gently fold together until almost combined.
Add the apple cider vinegar and fold through until no dry flour remains. Avoid overmixing, as this may make the muffins dense.
Divide the mixture evenly between the 12 muffin cases, filling each approximately three-quarters full.
Bake for 22–25 minutes, until the muffins are risen and a skewer inserted into the centre comes out clean or with a few moist crumbs.
Leave the muffins in the tin for 10 minutes, then transfer them to a wire rack and allow them to cool completely.
To make the butter icing, beat the softened dairy-free block butter for 2–3 minutes until smooth and creamy.
Add the icing sugar gradually, beating well after each addition.
Add the vanilla extract and beat for another 2–3 minutes until the icing is light and fluffy. If it is too stiff, add suitable dairy-free milk half a teaspoon at a time until it reaches the desired consistency.
Pipe or spread the butter icing over the completely cooled muffins.
Add your chosen suitable allergy-friendly decorations, or leave the icing plain.