Preheat the oven to 175 degrees Celsius. Grease a 8 x 4 inch rectangle tin and line the bottom with greaseproof paper.
Measure the flour, cocoa powder, baking powder and salt in a small bowl. Use a wire whisk and whisk the dry ingredients together until they are all well-mixed and evenly distributed.
Measure the dark chocolate and unsalted butter in a large heatproof bowl and melt them over a pot of simmering water, stirring occasionally.
Remove the heatproof bowl from the pot. Add in the caster sugar and whisk gently - not all of the sugar will dissolve, it's ok if the mixture is grainy. Let the mixture stand for 10 minutes to cool down slightly (the eggs will scramble if added to a hot mixture).
Crack in one egg into the chocolate mixture. Whisk the mixture (yes, same wire whisk!) for 30 seconds until it is fully incorporated. Whisk in the second egg, and then whisk in the vanilla extract.
Sift a third of flour mixture into the chocolate batter and folding gently using a large metal spoon or rubber spatula. until all the dry ingredients have been incorporated.
Transfer the cake batter into the lined cake tin. Bake for 25 to 30 minutes until the top bounces back when touched lightly and a skewer inserted into the centre of the cake comes out clean.
Cool the cakes completely on a wire rack. Serve with some whipped cream, fresh fruits, ice cream or even chocolate sauce!