Preheat oven to 350 degrees. Line a jelly roll (10x15”) pan with foil and spray with cooking spray (I like to use the spray with flour).
Beat eggs at high speed for 5 minutes, until frothy and dark yellow. Beat in sugar, oil, and vanilla extract.
Whisk together salt, baking powder, cinnamon, and flour. Stir 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 (mine took 10).
While the cake is baking, set a clean kitchen towel out on a large work surface. Sprinkle liberally with powdered sugar (about 1/4-1/2 cup). As soon as the cake comes out of the oven, turn it over on the kitchen towel sprinkled with powdered sugar. Remove foil carefully.
Working at the short end, fold the edge of the towel over the cake. Roll tightly, rolling up the cake into the towel. Let cool completely while rolled, at least two hours.
While the cake is cooling, 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. Chill until it firms up a bit, at least thirty minutes to one hour. Slice and serve.
Cake can be wrapped in plastic and frozen for up to one month.