Prep the pan and oven. Heat the oven to 350°F (175°C). Line a 10x15-inch jelly roll pan with parchment, leaving overhang on the long sides. Lightly grease the parchment. Lay out a clean kitchen towel and generously dust it with confectioners’ sugar.
Whip the eggs and sugar. In a large bowl, beat the eggs and granulated sugar with a hand mixer or stand mixer on high until very pale and tripled in volume, about 6–8 minutes. The mixture should fall in thick ribbons.
Add flavor and liquids. Gently mix in the oil (or melted butter), milk, vanilla extract, and vanilla bean seeds until just combined. Do not overmix.
Fold in dry ingredients. Sift the flour, baking powder, and salt over the egg mixture. Using a spatula, fold with light, sweeping motions until no dry streaks remain. Keep the batter airy.
Spread and bake. Pour the batter into the prepared pan and smooth the top. Tap the pan once to pop large bubbles. Bake 10–12 minutes, until the top springs back when touched and the edges are lightly golden.
Roll while warm. Immediately loosen the cake edges with a knife. Grasp the parchment and invert the cake onto the sugared towel. Peel off the parchment. Starting from the short end, roll the warm cake and towel together into a log. Let cool seam-side down on a rack for 45–60 minutes.
Make the filling. Beat the cream cheese until smooth. Add confectioners’ sugar, vanilla, and a pinch of salt; beat until creamy. In a separate bowl, whip the cold cream to medium peaks. Fold the whipped cream into the cream cheese mixture until smooth and fluffy.
Fill the cake. Carefully unroll the cooled cake. Spread the filling evenly, leaving a 1/2-inch border on the far short end. Sprinkle a small handful of sprinkles over the filling for a fun speckled look.
Roll and chill. Roll the cake back up without the towel, keeping it snug but not tight. Wrap in plastic and chill for at least 1 hour to set the shape.
Finish and serve. Unwrap, trim the ends for a clean look, and dust with confectioners’ sugar. Press sprinkles along the top or sides. Slice with a sharp serrated knife, wiping the blade between cuts.