This is a rich, dense chocolate cake that is almost flourless.
Preheat oven to 375 degrees. Coat a 12-by-17-inch rimmed baking sheet with cooking spray, and line with parchment paper. Coat parchment; dust with cocoa powder, tapping out excess, and set aside.
Sift flour, cocoa, cornstarch, and salt into a medium bowl; set aside. Add eggs and sugar to the bowl of an electric mixer. Place over a pan of simmering water. Whisk constantly until eggs feel hot to the touch (do not cook eggs). Transfer bowl to electric mixer; fit with the whisk attachment. Beat on high speed until pale and fluffy and the bowl is no longer hot, about 3 minutes. Remove bowl from mixer, and sift flour mixture over the top; fold in gently. When almost combined, pour melted butter down side of bowl; fold to combine. Pour onto prepared baking sheet, smoothing top.
Bake cake until it starts to pull away from sides and top is springy to the touch, about 10 minutes. Place a clean kitchen towel flat on work surface; dust with cocoa. Run a knife around edge of cake; invert onto towel. Remove parchment; starting from a long end, roll in towel. Transfer cake to a wire rack; let cool completely.
Unroll cake; soak with rum simple syrup using a pastry brush. Spread mousse evenly over cake, leaving a 1-inch border. Gently reroll cake. Transfer to the back of a rimmed baking sheet, seam side down; chill until mousse is firm, about 3 hours.
Frost cake with fudge frosting. Just before serving, cut into 3/4-inch-thick slices.