Prepare oven and baking sheet: Heat oven to 450 degrees, with rack in bottom half. Coat a 12 1/2-by-17 1/2-inch rimmed baking sheet with cooking spray. Line with parchment paper; coat with cooking spray. Dust with cocoa powder and tap out excess.
Combine dry ingredients: Whisk together cocoa powder, flour, and salt in a bowl.
Combine eggs and sugar: Bring about 2 inches of water to a simmer in a medium saucepan. In the heatproof bowl of an electric mixer set over (not in) simmering water, whisk eggs, yolks, and granulated sugar until sugar has dissolved and mixture is warm to the touch. Attach bowl to mixer fitted with whisk attachment; beat on medium-high speed for 2 minutes. Raise speed to high; beat until mixture is pale and thick, about 4 minutes more.
Add dry ingredients: Sift cocoa mixture over egg mixture; using a large rubber spatula, carefully fold. When almost incorporated, pour melted butter down side of bowl; gently fold to combine.
Bake: Using an offset spatula, spread batter evenly into prepared sheet. Bake cake until golden brown and springy to the touch, 6 to 7 minutes. Meanwhile, dust a clean kitchen towel with cocoa powder. Run a knife around sides of cake. Invert onto prepared towel, and remove parchment.
Roll into log and cool: While the sheet cake is still warm, gently roll, starting from one short side, into a log, incorporating the kitchen towel. Let the cake cool completely, rolled. (This will give the cake a "shape memory," so it will be easier to roll again with filling.)
Meanwhile, make rum syrup: In a small saucepan, combine the water and 1/4 cup sugar. Bring to a boil; reduce heat, and simmer until sugar has dissolved, about 3 minutes. Remove from heat; add rum, then let cool completely.
Fill cake and reroll: Unroll cake. Brush entire surface generously with cooled rum syrup. Whip cream and remaining tablespoon sugar until soft peaks form; spread over cake leaving a 1/2-inch border all around. Roll cake to enclose filling (without towel), starting at a short end. Wrap the towel around rolled cake, securing with clips or clothespins to help retain the shape.
Chill and serve: Refrigerate 30 minutes (or up to 3 hours). Dust with cocoa powder, cut into slices with a serrated knife, and serve.