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 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, and vanilla, then add flour and mix slowly until just combined.
Divide batter into three bowls. Color each as desired.
Pour each batter into prepared pan, spreading and swirling as desired 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 ¼ 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: Place the heavy whipping cream in a very large bowl or the bowl of a stand mixer. Add the pudding mix and beat at high speed with electric or hand mixer until stiff whipped cream forms. This will take anytime between 1-5 minutes, depending on the speed of your mixer. Taste, then add powdered sugar as desired for sweetness, whipping to mix them in.
To fill cake: Carefully unroll the cake. Spread with some of the whipped cream mixture. Carefully roll the cake back up as tight as possible, unsticking it from the towel as you go.
To decorate the cake: divide the remaining frosting into 2-3 bowls. Tint with frosting as desired. To frost as I’ve shown it: place each color in it’s own ziploc or piping bag, then cut of the tip. Pipe stripes of each color on the cake roll, then swirl with a spatula as desired. You can also use a 1M tip to make roses in different colors, however you like it (see this cake roll for an example of how that looks).
To decorate: go to town with sprinkles, brightly colored candies, marshmallows and more.
Chill for at least 1 hour before slicing. Cover loosely if chilling longer than one hour.