Preheat your oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners.
In a large bowl, whisk the gluten free flour, granulated sugar, brown sugar, baking powder, baking soda, cinnamon, nutmeg, and salt until well combined.
Make a well in the center.
Add the eggs, oil, applesauce, and vanilla. Whisk to combine, then switch to a spatula and fold until no dry streaks remain.
Fold in the grated carrots and any optional mix-ins. The batter will be thick but scoopable.
Divide the batter evenly among liners, filling each about 3/4 full.
Bake for 18–22 minutes, or until the tops spring back and a toothpick comes out clean or with a few moist crumbs.
Cool in the pan for 5 minutes, then transfer cupcakes to a wire rack to cool completely before frosting.
Make the frosting: Beat cream cheese and butter on medium speed until smooth and fluffy, 2–3 minutes.
Add vanilla and a pinch of salt.
Gradually add powdered sugar, beating on low to start, then medium until creamy and spreadable. Adjust with more sugar for a firmer frosting or a splash of milk for a softer one.
Frost cooled cupcakes with a knife or piping bag. Garnish with a sprinkle of cinnamon or chopped nuts if you like.