Preheat oven to 350°F. Line a 10x15” or 10.5x15” cake/jelly roll pan with foil and spray with floured nonstick cooking spray.
Place eggs in a large bowl or the bowl of an electric mixer. Beat at medium speed with mixer for 3-5 minutes until foamy and yellow. Add sugar and mix for 2 more minutes, until the mixture is thickened slightly. Mix in oil, baking powder, salt, vanilla, and cocoa, then add flour and mix slowly until just combined. Pour into prepared pan, spreading as needed with a spatula. Bake for 10-15 minutes, or until the top is browned and the cake springs back when touched lightly. (Mine took 12 minutes, but all ovens differ.)
While the cake is baking, lay out a clean kitchen towel onto the counter. Spread with about 1/4 cup powdered sugar.
Remove the hot cake from the oven and carefully, using oven mitts so you don’t burn yourself, flip the cake onto the towel. This might make a mess, but that’s okay. Carefully remove the pan and foil (they’re hot!) and then, using the towel, roll up the cake from the short side. The towel will be rolled into the cake. Let this cool completely before continuing.
Note: you can wrap the cooled cake in plastic wrap and let it sit overnight before finishing.
Make the filling: beat cream cheese and butter ntil creamy, then mix in powdered sugar slowly. Add vanilla and beat until smooth.
To fill cake: Carefully unroll the cake. Spread with the frosting. Carefully roll the cake back up as tight as possible, unsticking it from the towel as you go.
Chill for one hour before slicing. Before serving, dust the top with powdered sugar.
If the cake cracks, double the frosting recipe and frost the outside as well - no one will know!