Preheat oven to 350°F. Line a 10×15″ pan with foil and spray with cooking spray.
Combine together salt, baking powder, cinnamon, and flour. Set aside.
Beat eggs at high speed for 5 minutes, until frothy and dark yellow. Beat in sugar, oil, and vanilla extract.
Stir dry into wet ingredients just until blended. Stir in carrots.
Spread in prepared pan. Batter will be in a very thin layer and you will need to use a wooden spoon or spatula to spread it to all the corners of the pan.
Bake for 9-11 minutes.
While the cake is baking, set a clean kitchen towel out on a large work surface.
Sprinkle liberally with powdered sugar .
As soon as you remove the cake from the oven, turn it over on the kitchen towel sprinkled with powdered sugar. Remove foil gently.
Working at the short end, fold the edge of the towel over the cake.
Roll tightly, rolling up the cake in the towel.
Let cool completely while rolled, at least one hour.
Make the frosting.
Beat the butter and cream cheese together until smooth.
Beat in powdered sugar and vanilla.
When cake is cool, carefully unroll the towel. Spread the filling on the cake evenly, and re-roll tightly without the towel.