Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a bowl, whisk together flour, baking powder, baking soda, cinnamon, and salt.
In another bowl, beat eggs, granulated sugar, and brown sugar until thick. Add vanilla and carrots. Mix well.
Fold in the dry ingredients until just combined. Spread batter evenly in the prepared pan.
Bake for 12-15 minutes or until the cake springs back when touched. Let it cool for 5 minutes.
Turn the cake onto a towel dusted with powdered sugar. Roll up the cake in the towel and let it cool completely.
For the filling, beat cream cheese, powdered sugar, butter, and vanilla until smooth.
Unroll the cake, spread the filling evenly, and roll it back up. Chill before serving.