How to Make It
In a small nonreactive saucepan, whisk the egg yolks with the sugar and lemon juice. Cook over low heat, stirring constantly, until the mixture just begins to thicken; do no let it boil. Immediately strain the curd into a bowl. Place plastic wrap directly on the surface of the curd and refrigerate until cold.
Preheat the oven to 350°. Line a baking sheet with parchment or wax paper. Put the coconut in a medium bowl. In a cup, combine the sugar and boiling water; stir to dissolve the sugar. Add the sugared water to the coconut and toss until thoroughly coated. Spread the coconut evenly on the baking sheet and toast it in the oven for 6 to 10 minutes, stirring occasionally, until golden. Watch the coconut carefully so it doesn't burn. Transfer to a plate.
Preheat the oven to 400°. Grease a 15-by-10-inch jelly-roll pan and line it with wax paper.
Sift the flour and baking powder onto a sheet of wax paper. In a large heatproof bowl, whisk together the eggs, egg yolk and sugar. Set the bowl over (but not in) a pot of simmering water and whisk constantly until the mixture is warm to the touch. remove the bowl from the heat and beat the mixture on high speed until is is pale in color and as thick as whipped cream.
Meanwhile, in a small saucepan, melt the butter in the milk over low heat. Turn off the heat but leave the pan on the stove.
Sift one-third of the sifted flour over the egg mixture and gently fold it in with a large spatula; repeat in two more additions. Stir the milk mixture and pour half of it in a circle on the batter; fold it in. Repeat with the remaining milk and fold gently until incorporated. Pour the batter into the prepared pan and gently smooth the surface. Bake for 10 minutes, or until the cake springs back when gently pressed. Let the cake cool completely on a rack.
Put the egg whites in a grease-free metal bowl. In a small saucepan, stir the sugar with 1/3 cup of water and bring to a boil, wiping down the sides of the pan with a wet pastry brush. Boil the syrup until it reaches 236° on a candy thermometer, about 5 minutes.
Start beating the whites on medium speed until foamy, then add the cream of tartar and continue beating. When the sugar syrup reaches 242°, pour it onto the whites in a slow stream and continue to beat the frosting for 3 minutes, or until it is thick and fluffy; add the vanilla when almost finished. The frosting will still be slightly warm.
Turn the cooled cake out onto a piece of foil and peel off the wax paper. Place a piece of plastic wrap over the cake and flip it so that it is right side up again. Remove the foil.
Evenly spread the lemon curd over the cake. Spread about 2 1/4 cups of the marshmallow frosting over the curd. Starting at a long edge, roll the cake as tightly as possible and wrap it in plastic. Let stand at room temperature for about 5 minutes, seam side down.
Transfer the log to a platter and unwrap it. Spread the remaining frosting in a thick layer over the entire log. Scatter the toasted coconut over the log. Press with your hands to help the coconut adhere to the frosting. The cake can stand at room temperature for up to 4 hours. With a hot knife, slice about 1/2 inch off each end before serving.