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Preheat the oven to 180C/350F/Gas 4. Line a muffin tray with 9 paper cases.
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Sift the flour, cocoa powder and baking powder into a bowl. Add the sugar and butter and beat the mixture together until well combined.
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Whisk in the egg and milk until the mixture is thick and smooth.
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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.
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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.
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Spoon the icing onto the cupcakes, leaving a small border around the edge.
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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.
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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.)
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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.
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Set aside for 20 minutes to allow the icing to harden.