For the jam, put the strawberries in a pan and use a potato masher to squash the strawberries. Add the sugar. Cook on a medium heat until the jam reaches 105C or test for wrinkles on a cold plate to check if the jam is ready. Add the lime zest and juice and leave to cool. Once cooled, blend using a hand-blender to get rid of any lumps. Set aside.
For the chocolate ice cream, make sure the freezing components of your ice cream maker are in the freezer and frozen.
Warm the coconut cream in pan very gently - do not boil. Once warm, add the coffee granules.
In a bowl, whisk together the sugar, cocoa, salt and agar-agar. Add the eggs and mix well. Add the warmed coconut cream gradually, whisking all the time. Transfer the mixture back to the pan and stir for 8-10 minutes, or until the mixture has thickened. Once thick, transfer to a shallow tray and leave in the freezer for the mixture to cool. Once cooled, churn in the ice cream maker for at least 45 minutes. Place the ice cream in the freezer.
For the strawberry and lime mousse ice cream, whisk the coconut cream in a bowl until light and fluffy. Add the agave nectar, dried strawberries, 2 tablespoons of the strawberry jam, strawberries, agar-agar and lime zest and mix well. Whisk until the mixture is light and fluffy and well incorporated. Transfer to an ice cream maker and churn until soft set. Place in a piping bag and pipe a line about 2.5cm/1in wide and as long as your Swiss roll tin (about 33cm/13in). Wrap the piped line in cling film in the shape of sausage. Leave in the freezer to set completely.
For the decorative paste, mix all the ingredients, except the food colouring, together to a smooth paste. Add the food colouring to make a vibrant pink colour. Put the paste into piping bags fitted with a nozzle coupler and writing tip attached (ideally Wilton 2).
Draw a paisley template onto a sheet of baking parchment. Line a Swiss roll tin with the paper and grease slightly. Pipe the design using the decorative paste. Place the tray in the freezer for 10 minutes.
For the fatless sponge, preheat the oven to 200C/180C Fan/Gas 6. Whisk together the eggs and sugar in a bowl until they get to the ribbon stage (this means that the mixture holds its shape for a few seconds when you drizzle the mixture back over itself). Sift the flour and fold it in with a metal spoon. Spread the mixture over the design and bake for 8-10 minutes on the middle shelf.
Remove the sponge from the oven and turn out onto a sheet of baking parchment on a cooling rack. Press the sponge down and trim to make slightly thinner if need be. Turn it over so that the pattern side is facing down then roll up gently and leave to cool in a clean tea towel.
For the dipped strawberries, break the chocolate into pieces and melt in separate bowls set over a pan of simmering water. Place dots of white chocolate into the dark chocolate and drag a skewer through to feather it. Dip the strawberries in the feathered chocolate and leave to set in the fridge.
To assemble, pipe the softened chocolate ice cream onto a sheet of cling film into a rectangle as long as the Swiss roll tin and approximately 12.5cm/5in wide. Place the frozen strawberry mousse on top. Wrap in cling film encasing the strawberry mousse into the chocolate ice cream. Roll into cling film and leave in the freezer to firm up again. Unroll the sponge and layer with a thin spreading of the jam.
Place the firm ice cream roll on top and roll together. Cut off any uneven ends, re-wrap in greaseproof paper and return to the freezer to freeze.
Squeeze to make everything adhere. Unwrap and place on a serving platter. Arrange the dipped strawberries on top. Scatter over some dehydrated strawberries and serve.