Preheat your oven to 350°F (175°C). Line a 10×15-inch jelly roll pan with parchment paper and lightly grease it.
In a bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In another bowl, beat the eggs, granulated sugar, and brown sugar until they are pale and fluffy. Stir in the vanilla extract.
Gently fold in the dry ingredients, then add the grated carrots. Spread the batter evenly into the prepared pan.
Bake for 10–12 minutes, or until the top springs back when lightly touched.
While the cake is still warm, carefully roll it up (with parchment paper) from the short end. Let it cool completely while rolled up.
Beat the cream cheese, butter, powdered sugar, and vanilla extract until smooth and creamy.
Unroll the cooled cake and spread the cream cheese filling evenly. Gently re-roll the cake without the parchment paper this time.
Chill the cake roll in the refrigerator for at least 1 hour to set. Dust with powdered sugar before serving, and garnish with chopped nuts if desired.