Heat the oven to 180°C/Fan 160°C/Gas 4. Brush two 20cm loose-bottomed cake tins with oil and line with parchment. To make the sponge, place the sugar and eggs in a mixing bowl and whisk until they are light in colour and frothy. Add the remaining ingredients. Stir until combined and pour into the cake tins.
Bake for 20–25 minutes until golden brown and the sponge is just coming away from the edge of the tins. Set aside to cool for 5 minutes, then run a palette knife around the inside edge of the tins. Carefully turn the cakes out onto a cooling rack to cool completely.
To make the blackberry compote, put the blackberries, sugar and lemon juice in a saucepan. Bring to a simmer and cook over a low–medium heat for 10 minutes or until the blackberries have broken down. Push through a sieve and return to a clean saucepan. Mix the arrowroot with 1 teaspoon water to form a paste. Stir into the purée and cook over a medium heat, stirring all the time until just simmering. Set aside to cool – it will thicken slightly as it cools.
Place the ingredients for the buttercream in a large bowl or the bowl of an electric mixer. Set your mixer to a low speed and gradually increase it until you have a smooth, creamy buttercream.
To assemble, place one cake upside down and spread it with some buttercream. Turn the other cake upside down and spread a generous layer of blackberry compote. Sandwich together. Spread a thick layer of buttercream on the top and sides of the cake, smoothing the edges with a palette knife (for a super smooth finish, dip the knife into hot water before you spread). Finish by swirling some compote through the buttercream on the top of the cake.