Pre-heat your oven to 190°c (170°c for fan-assisted ovens or Gas Mark 5) and line a (approx) 35x25cm Swiss roll tin with grease proof paper and grease with butter, then dust with flour.
In a bowl, sift the flour and cocoa together and set aside.
In a large, grease-free bowl add the eggs and whisk lightly before adding the sugar. Whisk together until you have a light coloured, creamy, thick batter. This can take a while by hand so an electric mixer is preferable.
Add the sifted flour/cocoa an gently fold to create a batter. Add a tbsp of water and fold in also.
Pour the batter into your prepared tin and spread over evenly. Bake in your pre-heated oven for 8-10 minutes.
Have a piece of greaseproof paper ready on your work surface. Turn out the baked sponge onto the greaseproof paper. Whilst still hot, peel off the greaseproof paper on the sponge (from the tin). Trim a thin strip from the 4 sides of the sponge, to remove the crisp edges.
On one narrow end, score the sponge about 1cm from the edge with a knife, being careful not to cut right through. Starting on this edge, roll the sponge up in the greaseproof paper. Leave to cool.
Prepare your buttercream by creaming the butter and icing sugar until light and fluffy, before adding the cocoa, vanilla and milk. Mix until combined. If your buttercream feels a little thick, add another tbsp of milk.
Once cool, unroll your sponge and spread the inside with about 1/3 of the buttercream. Re-roll the sponge, leaving behind the greaseproof paper and sit the sponge seam down.
Cut a section off at an angle and attach to the side with some buttercream to create the log shape. Spread the sponge with the remaining buttercream, finishing with a fork for a bark-like effect. Decorate as desired with things like fondant holly leafs, pine cones and even a robin. Dust with icing sugar before serving.