Preheat the oven to 350°F (175°C) and line a 12-cup muffin tin with paper liners.
In a medium bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, and nutmeg.
In a large bowl, whisk eggs, granulated sugar, and brown sugar until combined. Add oil and vanilla; whisk until smooth.
Fold the dry ingredients into the wet ingredients, then stir in grated carrots until just combined.
Divide batter evenly among the muffin cups. Bake for 22 to 25 minutes, or until a toothpick comes out clean.
Cool cupcakes in the pan for 5 minutes, then transfer to a wire rack to cool completely before frosting.