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, and vanilla, then add flour and mix slowly until just combined. Stir in sprinkles. 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: Melt the chocolate in a microwave safe bowl for about 2 minutes, stirring every 30 seconds, until the chocolate is melted and smooth. Cool 10 minutes. While the chocolate is cooling, beat the butter with a hand mixer for about 1 minute until it’s totally creamy. Add the powdered sugar and mix until fluffy, then mix in vanilla and salt. Beat in the cooled chocolate until smooth. Use immediately, or store covered at room temperature for up to 3 hours.
To fill cake: Carefully unroll the cake. Spread with some of the frosting. Carefully roll the cake back up as tight as possible, unsticking it from the towel as you go. Frost with remaining frosting and sprinkle with sprinkles.
You can serve immediately or chill until ready to serve. Cover with plastic wrap if storing longer than 2-3 hours.