Preheat the oven to 180C/350F/Gas 4. Line a muffin tray with 9 paper cases.
Sift the flour, cocoa powder and baking powder into a bowl. Add the sugar and butter and beat the mixture together until well combined.
Whisk in the egg and milk until the mixture is thick and smooth.
Divide the mixture between the paper case and bake in the oven for 20-25 minutes, or until springy to the touch and skewer inserted into the middle of one of the cakes comes out clean. Set aside to cool on a wire rack.
For the cream cheese icing, beat the cream cheese and butter together in a bowl until well combined. Beat in the icing sugar and vanilla.
Spoon the icing onto the cupcakes, leaving a small border around the edge.
For the chocolate icing, sift the icing sugar and cocoa powder into a bowl. Stir in 2 tablespoons of water to form a smooth, thick icing.
Transfer the mixture to a piping bag with a fine nozzle. (Alternatively, spoon the mixture into the corner of a sandwich bag and snip off the corner, to form a small hole.)
Carefully pipe 3 concentric circles onto each cupcake. Run a toothpick from the centre to the edge of the cake, through each circle of icing, at 2cm/1in intervals to create a cobweb effect.
Set aside for 20 minutes to allow the icing to harden.