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Up to a month before, put all the soup ingredients in a pan and bring to a simmer. Continue to simmer, covered, for about 20-30 mins or until the artichokes and potatoes have become very tender. Liquidise the soup in batches until completely smooth. If you want to thin it, add more stock. Pour the soup into a rigid plastic container and leave to cool before covering and freezing.
The day before serving, transfer the soup to the fridge to defrost overnight.
On the day, heat oven to 190C/fan 170C/gas 5. Heat the soup and check the seasoning. While the soup is warming, put the grissini sticks on a baking tray. Scatter them with 1 tbsp of the Parmesan and put in the oven until warm and the cheese has melted, about 8-10 mins.
Meanwhile, pour the cream into a small pan and simmer to reduce by a third. Stir in remaining Parmesan and let it melt. (For a little extra body, add a tbsp or two of soup.) Stir in 1-2 tbsp milk to cool it down, then blitz with a stick blender, if you like, for a frothy finish. (To avoid any splashes, transfer the cream to a larger pan or deep bowl to do this.) Ladle the soup between soup bowls, topping each with the warmed cream and a sprinkling of chives if using. Serve with the warm grissini sticks.