Make the cake: Preheat the oven to 350 degrees F. Butter an 11-by-17-inch rimmed baking sheet and line the bottom with parchment paper. Lightly butter the parchment and dust with flour, tapping out the excess.
Whisk 1 cup each flour and granulated sugar, the baking powder, baking soda and salt in a medium bowl. In a large bowl, whisk the 4 egg yolks, vegetable oil, coconut milk and coconut extract. Whisk in the flour mixture in 2 batches.
Beat the 6 egg whites and cream of tartar in a large bowl with a mixer on medium speed until foamy. Increase the speed to medium high; gradually add the remaining 1/2 cup granulated sugar and beat until stiff, shiny peaks form. Fold one-third of the beaten egg whites into the egg yolk mixture with a rubber spatula, then gently fold in the rest.
Spread the batter on the prepared baking sheet and bake until the cake is golden and springs back when pressed, 15 to 20 minutes. Loosen the edges of the cake with a knife; transfer the baking sheet to a rack and let the cake cool slightly. Dust a large piece of parchment paper with confectioners' sugar. Invert the cake onto the parchment and peel off the parchment on top. Brush the cake with the rum.
Make the frosting: Whisk the egg whites and granulated sugar in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water) until the mixture is warm and the sugar dissolves. Remove the bowl from the pan; cool slightly.
Beat the egg white mixture with a mixer on medium-high speed until stiff peaks form, 12 to 15 minutes. Beat in the butter a few pieces at a time, then continue beating until glossy and smooth, 5 to 7 more minutes. (Don't worry if the mixture looks curdled at first; continue beating.) Beat in the rum.
Spread about 2 cups of the frosting on the cake, then sprinkle with 2 cups coconut. Starting at a short end, roll up the cake, using the parchment to help you. Transfer to a platter, seam-side down. Cover the cake roll with the remaining frosting and sprinkle with the remaining 2 cups coconut.
Photograph by Johnny Miller
Recipe courtesy Food Network Magazine