Preheat the oven to 220C/425F/gas mark 7. Grease a 23x33cm/9x13in swiss roll tin and line with baking parchment.
Beat the eggs and sugar together in a large bowl with an electric whisk until they are extremely pale, mousse-like and at least tripled in volume. If you lift the beaters out of the bowl and wiggle them, the mixture that runs off should leave a trail on the surface of the bowl that disappears after a couple of seconds.
Sift the flour into the bowl and gently fold it in, being careful not to knock out the air. Pour the mixture into the prepared tin and tilt it so that the batter runs into the corners. Bake for about 10 minutes until it is lightly browned, springy and just starting to pull away from the edges.
While the sponge is baking, cut a piece of baking parchment slightly larger than the size of the tin and sprinkle it with caster sugar. Carefully flip the (hot) baked sponge out onto the parchment and peel the baking parchment off the bottom.
Score the sponge 2.5cm in from one of the short edges, being careful not to cut all the way through. Tightly roll the sponge up from the scored edge with the clean baking parchment inside and set aside until cooled completely.
While the sponge is cooling make the buttercream. Place the egg whites and sugar in a spotlessly clean heatproof bowl (the bowl of your stand mixer if you have one) and place the bowl over a pan of gently simmering water, make sure that the base of the bowl doesn't touch the water. Gently whisk the eggs (with a hand whisk, not an electric one) until the sugar has dissolved, if you rub the mixture between your fingers it should feel smooth with no hint of graininess. It should measure about 60C/140F on a thermometer.
Remove the bowl from the heat and whisk with an electric mixer/stand mixer on high until the egg whites have formed a stiff, glossy meringue and the mixture is completely cold, the base of the bowl should feel cool to the touch, this will take about 10 minutes.
Turn the mixer down to low and gradually add the butter, a tablespoon at a time, mixing to incorporate after each addition. The mixture may curdle or begin to look soupy, just keep on whisking and it will come back together.
Once all of the butter has been incorporated and the buttercream is smooth and silky, gradually whisk in the coconut milk, a little at a time, then the vanilla extract and salt.
Once the sponge is completely cold, carefully unroll it and spread with the jam, leaving about a centimeter border. Spread about two thirds of the buttercream over the top then re-roll the sponge.
Spread the remaining buttercream over the top and sides of the cake then press the coconut onto the buttercream using your hands; trim the ends of the swiss roll to neaten then serve.
The cake is best served on the day it is made but it will keep for a day or so in an airtight container.