Prep your pan and oven: Heat oven to 375°F (190°C).
Line a 10x15-inch (25x38 cm) jelly roll pan with parchment, leaving overhang on the long sides. Lightly grease the parchment.
Mix dry ingredients: In a bowl, whisk cocoa, flour, baking powder, and salt until combined. Sift if lumpy.
Set aside.
Whip eggs and sugar: In a large bowl, beat eggs, sugar, and vanilla on high speed for 5–7 minutes until pale, thick, and tripled in volume. The mixture should ribbon off the whisk.
Fold in dry mix: Sift the dry ingredients over the egg mixture in two additions. Gently fold with a spatula, turning the bowl and cutting through the center to keep the batter airy.
Add milk and oil: In a small cup, whisk milk and oil.
Stir a spoonful of batter into it, then fold this mixture back into the main batter just until blended. Do not overmix.
Spread and bake: Pour batter into the pan and smooth to the corners. Tap the pan once to release large bubbles.
Bake 9–12 minutes until the top springs back and a toothpick comes out with a few moist crumbs.
Prepare for rolling: While the cake bakes, lay a clean kitchen towel on the counter and dust it with powdered sugar or cocoa. You can also use a fresh sheet of parchment.
Turn out the cake: Remove the cake and run a knife around the edges. Invert it onto the sugared towel or parchment.
Peel off the baking parchment carefully.
Roll while warm: Starting from a short side, roll the cake up with the towel or parchment inside. Roll gently but snugly. Let it cool completely on a rack, seam-side down, about 45–60 minutes.
Make the filling: In a chilled bowl, beat mascarpone (or cream cheese) with powdered sugar and vanilla until smooth, 15–30 seconds.
Add cold heavy cream and whip to medium-stiff peaks. Do not overwhip.
Unroll and fill: Carefully unroll the cooled cake. Spread the cream evenly, leaving a 1/2-inch border on the far edge to avoid overflow.
Reroll and chill: Roll the cake back up without the towel.
Wrap in parchment or plastic and chill for 30–60 minutes to set.
Finish and serve: Trim the ends for clean edges. Dust with cocoa or powdered sugar, or drizzle with a thin chocolate glaze. Slice with a sharp, serrated knife, wiping the blade between cuts.