Beat the butter and sugar until pale and fluffy. Whisk in half the beaten egg (you only need half) and the vanilla extract, then stir in the flour and cocoa powder until just combined.
Divide the mixture in half, shape each into balls, wrap both in clingfilm, then chill for a couple of hours until the dough has firmed up.
Preheat the oven to 180C°/350°F/gas 4, and grease and line a baking sheet with greaseproof paper.
Dust a work surface with icing sugar and roll out 1 batch of dough to 5mm thick. Cut out rounds using a 5cm cutter (re-roll and cut any trimmings), then transfer the biscuits to the baking sheet.
Bake for 10 minutes till cooked through. Cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely. Repeat with the second batch of dough.
To make the peppermint meringue filling, place the sugar, egg whites, cream of tartar and 1½ tablespoons of water in a large heatproof bowl. Beat with an electric whisk till fluffed up, about 1 minute. Place the bowl over a pan of simmering water set over the lowest heat possible, and continue to beat at high speed for 10 to 12 minutes, until stiff peaks form. Remove from the heat, add a few drops of peppermint extract and beat for 2 more minutes, until the meringue has thickened even more.
Fit a 1cm plain nozzle to a piping bag and fill the bag with the peppermint meringue. Pipe onto the biscuits in an upward spiral, working from the outside in; leave a small border around the edge – aim for a swirl around 3-4cm tall. Place the biscuits on a tray and leave to set in the fridge while you make the chocolate topping.
Place the chocolate pieces and oil in a small heatproof bowl and gently melt over a pan of simmering water until completely smooth and glossy. Transfer to a small, deep bowl and leave to cool at room temperature for about 10 minutes.
To finish the teacakes, hold the chilled biscuits by the base and gently dip the meringue into the melted chocolate, swirling so it’s completely covered. Or, place the biscuits on a wire rack with greaseproof paper underneath and drizzle over the chocolate until each biscuit is covered. Chill the biscuits for a few hours until the chocolate has completely set, and serve.