Preheat your oven to 350°F (175°C). Line a muffin tin with paper liners.
In a large bowl, mix the grated carrots and chopped walnuts.
In another bowl, whisk together the gluten-free flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a separate bowl, beat the eggs and sugar until well combined. Add the vegetable oil and vanilla extract, and mix well.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Fold in the carrot and walnut mixture.
Spoon the batter into the prepared muffin tin, filling each cup about ⅔ full.
Bake for 20-25 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Allow the muffins to cool in the tin for 5 minutes, then transfer to a wire rack to cool completely.