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Heat the oven to 200C/180C fan/gas 6, and line a baking tray with baking parchment. Beat the butter and both sugars in a large bowl using an electric whisk for 2 mins until smooth and fluffy.
Add the vanilla and beat until combined, scraping down the sides once. Add the flour, cream of tartar, bicarbonate of soda and ¼ tsp salt, and mix until just combined. Stir in the milk to loosen the mixture.
For the topping, combine the sugar and cinnamon in a shallow bowl. Roll 40g of the dough into a ball. Roll the dough ball in the cinnamon-sugar mixture, then place it on the prepared tray and press it down lightly. Repeat with the remaining dough and topping, spacing the balls about 5cm apart.
Bake for 10-12 mins until golden and puffed up. Leave to cool on the tray for 5 mins, then transfer to a wire rack and leave to cool completely. Will keep in an airtight tin for up to four days.