1. Melt the butter in the microwave or gently on the hob in a small pan. Set to one side. Preheat your oven to 180°C/Fan 160°C/Gas 4. Line a shallow rectangular tin, approximately 40cm x 28cm, with baking paper.
2. Separate the eggs, sliding the yolks into a medium-sized mixing bowl and the whites into a separate, dry mixing bowl (ideally, not plastic). Using a hand whisk, electric mixer or stand mixer, beat the egg whites till they form soft peaks.
3. Add the sugar and vanilla essence to the egg yolks. Using the whisk or the mixer you used for the egg whites, beat them till they're pale and they have increased in volume.
4. Sift the flour into the egg yolk mix with a pinch of salt, then gently fold together. Fold in 2 tbsp of the whipped egg whites to loosen the mix, then add the remaining egg whites, folding gently to keep in as much air as possible.
5. Pour the mixture into the lined tin and gently tip the mix to cover the paper evenly. Use a spatula to tease it to the edges, if you need to, but use a light touch. Slide the tin into the oven and bake for 12-14 mins till golden and springy when lightly pressed.
6. While the Swiss roll sponge is baking, finely zest the limes into a small bowl. Add the 3 tbsp sugar and mix. Cut a rectangle of baking paper, slightly larger than the cake tin, and sprinkle the lime sugar over it.
7. Remove the sponge from the oven. Carefully lift the sponge from the tin while warm, holding the edges of the baking paper. Confidently flip the sponge on to the sugar dusted paper. Peel off the baking paper from the top of the sponge. Trim all the edges to neaten, if you like. Turn the sponge so a short edge is nearest to you then score a line 1cm across this edge – this will help you roll the sponge tightly. While the sponge is still warm, roll up the Swiss roll as tightly as you can, rolling the paper inside with it. Leave it rolled up till completely cool.
8. While the Swiss roll is cooling, spoon the mascarpone into a large bowl and beat a couple of times to loosen it. Spoon in the gooseberry jam and gently fold it through to create ripples.
9. Gently unroll the Swiss roll – it might crack a little, but don't worry as the jam and mascarpone will glue it together. Spread the mascarpone and jam mix onto the Swiss roll, leaving a 2cm border. Roll the Swiss roll back up, minus the paper. The outside should be dusted with lime sugar.
10. Place the Swiss roll on a plate, seam side down, and serve in slices. The Swiss roll will keep for 1-2 days in the fridge.