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Heat the oven to 190C/170C fan/ gas 5. Butter and line a 20cm square cake tin with baking parchment.
Beat the butter, sugar and tahini together until pale and smooth. Beat in the eggs a little at a time, then add the vanilla, scraping down the sides of the bowl as needed.
Mix the flour, bicarbonate of soda and ½ tsp salt together, then fold into the butter mixture. Stir in the milk. Carefully fold in the banana and sesame seeds, then spoon the batter into the prepared tin and bake for 40-50 mins until a skewer inserted in the middle comes out clean. Leave to cool slightly in the tin, then lift out and transfer to a wire rack to cool completely.
To make the buttercream topping, beat all the ingredients, except the chocolate chips, together until smooth. Spread this over the cooled cake, then scatter over the chocolate chips and cut the cake into squares or rectangles to serve.