Preheat oven to 180°C (355°F). Grease a 20cm (8in) round springform cake tin and line the bottom and sides with baking paper.
Cut butter into small cubes and break chocolate into small pieces. Place the butter and chocolate into a small saucepan over low heat, stirring until melted, glossy and smooth. Remove from heat and set aside to cool for 10 minutes.
Beat eggs, caster sugar and brown sugar in a large mixing bowl until thick, pale and creamy. Stir in the cooled melted chocolate mixture along with the vanilla extract. Sift the flour and cocoa into the bowl and mix everything together until combined. Pour the brownie batter into the prepared tin and bake in the preheated oven for 30-40 minutes until cooked but still slightly tacky in the centre.
Remove from the oven and set aside to cool completely in the tin. Once cooled, use your fingers to lightly press the edges of the brownie down slightly so the brownie is nice and level and pushed up to the edges of the tin ensuring there is no gap between the brownie and the tin (see notes). The brownie must be completely cooled down before proceeding to the next step.
Remove the ice-cream from the freezer and allow it to defrost slightly (approximately 15-20 minutes). You want the ice-cream to be partially defrosted - you don’t want to melt the ice-cream completely. Tip the partially melted ice-cream into a bowl and use a spoon to mix and combine the more melted parts into the more frozen parts to achieve a spreadable but still thick consistency. Once mixed together, spoon or tip the ice-cream onto the brownie and carefully spread it out to form an even layer. Place in the freezer for a minimum of 4-5 hours or overnight.
The cake will keep in the freezer for 10 days so can be made in advance.