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1
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.
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2
Lightly butter the parchment and dust with flour, tapping out the excess.
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3
Whisk 1 cup each flour and granulated sugar, the baking powder, baking soda and salt in a medium bowl.
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4
In a large bowl, whisk the 4 egg yolks, vegetable oil, coconut milk and coconut extract.
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5
Whisk in the flour mixture in 2 batches.
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6
Beat the 6 egg whites and cream of tartar in a large bowl with a mixer on medium speed until foamy.
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7
Increase the speed to medium high; gradually add the remaining 1/2 cup granulated sugar and beat until stiff, shiny peaks form.
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8
Fold one-third of the beaten egg whites into the egg yolk mixture with a rubber spatula, then gently fold in the rest.
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9
Spread the batter on the prepared baking sheet and bake until the cake is golden and springs back when pressed, 15 to 20 minutes.
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10
Loosen the edges of the cake with a knife; transfer the baking sheet to a rack and let the cake cool slightly.
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11
Dust a large piece of parchment paper with confectioners' sugar.
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12
Invert the cake onto the parchment and peel off the parchment on top.
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13
Brush the cake with the rum.
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14
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.
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15
Remove the bowl from the pan; cool slightly.
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16
Beat the egg white mixture with a mixer on medium-high speed until stiff peaks form, 12 to 15 minutes.
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17
Beat in the butter a few pieces at a time, then continue beating until glossy and smooth, 5 to 7 more minutes.
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18
(Don't worry if the mixture looks curdled at first; continue beating.)
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19
Beat in the rum.
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20
Spread about 2 cups of the frosting on the cake, then sprinkle with 2 cups coconut.
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21
Starting at a short end, roll up the cake, using the parchment to help you.
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22
Transfer to a platter, seam-side down.
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23
Cover the cake roll with the remaining frosting and sprinkle with the remaining 2 cups coconut.
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24
Photograph by Johnny Miller
