Preheat oven to 350°F/175°C. Lightly grease the bottom of a 10 x 15-inch jelly roll pan. Cut parchment or foil to fit the bottom of the pan, then grease the paper or foil and the sides of the pan.
In a large bowl, beat the eggs at high speed for about 5 minutes. Then beat in the sugar, oil, and vanilla.
In a small or medium bowl, whisk together the salt, baking powder, ginger, nutmeg, cinnamon, and flour. Stir the dry ingredients into the wet, just until blended. Then stir in the carrots. Spread batter evenly into the prepared pan. Bake for 9-11 minutes. Edges should be set and the center of the cake should spring back when you lightly press on it.
While the cake is baking, lay a clean kitchen towel out on a work surface. Sprinkle about 1/4 cup powdered sugar onto the towel covering an area about the size of the cake. As soon as you take the cake out of the oven, turn it out onto the prepared towel. Remove the pan and carefully peel off the foil or parchment.
On one of the short ends, fold the edge of the towel over the cake, then roll the cake and the towel up all the way. Let cool for at least 1 hour.
Once the cake is cooled, make the filling. Beat the cream cheese until smooth, then beat in the melted butter and vanilla. Slowly mix in the powdered sugar and beat until smooth.
Gently unroll the cake. Spread the filling evenly over the cake and carefully re-roll the cake tightly. You can use the towel to help keep your fingers from sticking to the cake.
Chill cake until filling is set, at least 30 minutes to 1 hour. Dust with powdered sugar, then slice and serve.
Cake can be wrapped in plastic and frozen for up to 1 month.