Make the cake: Preheat the oven to 350 degrees F. Brush an 11-by-17-inch rimmed baking sheet or jelly roll pan with coconut oil; line the bottom with parchment paper. Brush the parchment with more oil and dust with flour, tapping out the excess.
Sift the flour, baking powder and salt into a large bowl; set aside. Place the eggs (in their shells) in a medium bowl and cover with hot tap water; let stand 5 minutes, then drain. Crack the eggs into a separate large bowl and add the granulated sugar. Beat with a mixer on medium-low speed, 30 seconds, then increase the speed to medium high and beat until thick, pale and fluffy, 8 to 10 minutes. (Don't underbeat; the beaten eggs are essential to help the cake rise.) Pour the egg mixture over the flour mixture and gently fold together with a rubber spatula until combined. Add the melted coconut oil and coconut extract and stir to just combine. Spread the batter in the prepared pan; bake until the cake springs back when gently pressed, about 18 minutes. Let cool 5 minutes.
Generously sift confectioners' sugar over the top of the warm cake and loosen the edges with a knife; lay a clean kitchen towel (not terry cloth) over the cake. Place another baking sheet over the towel and flip the cake with the baking sheets to invert it onto the towel. Remove the top baking sheet and parchment. Starting at a short end, roll up the cake and the towel together into a spiral. Transfer seam-side down to a rack and let cool completely.
Meanwhile, make the frosting: Spread the coconut on a baking sheet and bake at 350 degrees F, stirring once, until lightly browned and toasted, 5 to 7 minutes. Let cool. Whisk the egg whites, granulated sugar, lemon juice and salt in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water). Cook, whisking, until the sugar dissolves, 2 to 3 minutes. Remove the bowl from the heat and beat the egg white mixture with a mixer on medium-high speed until thick, glossy and cooled, about 6 minutes. Beat in the vanilla and coconut extracts.
Gently unroll the cake. Spread the jelly over the cake about 1/4 inch from the edges; reroll (without the towel). Trim the ends with a serrated knife. Transfer to a platter and spread the frosting on the top and sides of the cake. Sprinkle with the toasted coconut.
Photograph by Con Poulos