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To make the ice cream, whisk the egg whites in a bowl with an electric whisk until stiff. In another bowl, whisk the yolks, icing sugar and vanilla seeds until pale and billowy. In a third bowl, whip the double cream until holding its shape.
Fold the yolk mixture into the cream, then fold in the egg whites. Pour into a 900g loaf tin lined with a sheet of baking parchment bigger than you’ll need – this will help you later. Freeze for 2 hrs.
Remove the ice cream from the freezer and tin when it feels set but soft. Use a spatula to shape the mixture into a rough sausage shape, then use the parchment to roll the mixture tightly together into a cylinder about 35cm in length.
Twist the ends of the baking parchment like a cracker and return to the freezer for a further 1 hr.
To make the jam, first put a saucer in the freezer. Put the strawberries, lemon juice and sugar in a heavy- based saucepan over a high heat. Bring to the boil, stirring occasionally. Boil rapidly for 5 mins, then take off the heat.
Put 1/4 tsp of the jam onto the cold saucer – if it wrinkles when pushed with your finger, the jam is ready. If it’s still runny, return it to the heat and boil again for another 2-3 mins, then test again. Once the jam is set, leave it to cool.
Heat oven to 200C/180C fan/gas 6. Line a 24 x 32cm Swiss roll tin with baking parchment (with plenty of excess) and grease well. Whisk the eggs and sugar together with an electric hand whisk for 2-3 mins or until thickened and pale.
Fold the flour, baking powder and vanilla extract into the egg mixture with a large metal spoon until there are no pockets of flour visible.
Gently tip the mixture into your prepared Swiss roll tin and bake in the oven for 15 mins or until lightly golden and springy to the touch.
Remove from the oven, allow to cool for 1-2 mins or until cool enough to handle, then carefully roll up the sponge while it’s still warm (keeping the baking parchment attached). Leave to cool completely in its rolled-up shape.
Uncurl the sponge and spread the jam on the inside. Working quickly, unwrap the ice cream and put it on top of the jam. Roll the cake up around it.
Wrap the Arctic roll in a new sheet of baking parchment and a layer of foil too. Return to the freezer for a further 20-30mins, then unwrap, slice and serve immediately.