Preheat the oven to 180°C (160°C fan) and lightly grease a deep 8-inch round cake tin, lining the base with parchment paper.
To make the cake place the butter and sugar into a large bowl and use the Microplane Zester to grate in the zest of the lime. Beat with an electric mixer for about 5 minutes or until light and fluffy. Add the egg a little at a time beating until fully combined before adding more. Mix together the remaining ingredients and add to the butter mixture, using a spatula to fold together until fully combined. Scrape the batter into the prepared pan and gently level out. Bake in the preheated oven for 45-50 minutes or until coming away from the sides of the tin. Allow the cake to cool for 10 minutes before turning out onto a wire rack to cool completely.
For the buttercream place the egg white and sugar into small bowl set over a pan of simmering water and whisk gently until the mixture is warm to the touch and the sugar has dissolved. Use the Microplane Zester to grate in the zest of one of the limes and then use an electric hand mixer to whisk on high speed until the meringue is cool to the touch. With the mixer still running add the butter a little bit at a time whisking until you have a smooth buttercream.
To decorate spread the jam over the top of the cake. Place the buttercream into a piping bag fitted with a plain round tip and pipe the buttercream into rounds on top of the jam. Using the Microplane coarse grater top the cake with the zest from the remaining lime and top with the blackberries.
Kept in a sealed container this cake will keep for up to four days.