Preheat the oven to 350°F (175°C). Line a 10×15" jelly-roll pan with foil and spray with nonstick baking spray that contains flour, or grease and flour the foil. (If you prefer parchment, line the pan with parchment and spray the parchment with cooking spray.)
In a small bowl, whisk together the 1 teaspoon baking powder, ¼ teaspoon salt, and ¾ cup (93 g) all-purpose flour. Set aside.
In the bowl of a stand mixer or a large bowl with a hand mixer, beat the 3 large eggs on high speed for 5 minutes, until frothy and pale/dark yellow.
With the mixer running, add the ¾ cup (150 g) granulated sugar, 1 tablespoon (15 ml) vegetable oil, and 1 teaspoon vanilla extract. Beat just until combined.
Gently fold the dry flour mixture into the egg mixture with a spatula or wooden spoon, stirring only until just blended. Do not overmix.
Pour and spread the batter evenly into the prepared pan. The batter will be very thin—use a spatula or the back of a spoon to spread it into all corners.
Bake for 10–17 minutes, until the cake springs back when lightly touched and a toothpick inserted in the center comes out clean.
While the cake bakes, lay a clean kitchen towel on a large work surface and sprinkle it liberally with powdered sugar (powdered sugar for rolling, from the ingredient list) to prevent sticking.
As soon as the cake comes out of the oven, immediately invert the hot cake onto the prepared towel. Carefully remove the foil or parchment.
Starting at the short end of the cake, fold the edge of the towel over the cake and roll the cake and towel together tightly into a spiral. Place the rolled cake seam-side down and let it cool completely while rolled, at least 1 hour.
While the cake cools, make the frosting: beat the ½ cup (113 g) softened unsalted butter until fluffy. Add the 2 cups powdered sugar and 1 teaspoon vanilla extract and beat until smooth. Add 1–2 tablespoons heavy cream, a little at a time, until you reach a spreadable/piping consistency.
Divide the frosting into two bowls. Tint one portion blue and the other portion red with gel food coloring (red and blue gel food coloring from the ingredient list). Transfer each colored frosting into its own zip-top bag and seal; set aside.
When the cake is completely cool, unwrap and carefully unroll the towel from the cake. Peel the towel off.
Snip a small corner from the bag with the blue frosting and pipe evenly spaced horizontal stripes along the length of the unrolled cake. Use a knife or offset spatula to slightly smooth each stripe. Repeat with the red frosting, alternating stripes as desired.
Re-roll the cake tightly (without the towel), wrap it in plastic wrap, and chill until the frosting firms, at least 30 minutes to 1 hour. The cake may also be wrapped and frozen for up to one month.
Before serving, top the chilled roll with 8 ounces Cool Whip (or 2–3 cups freshly whipped cream) and add sprinkles. Lightly dust with powdered sugar (powdered sugar for dusting, from the ingredient list) if desired. Slice and serve.