Put the kettle on the boil. Don an apron; cranberry stains are stubborn.
Pick through your cranberries for rotten ones and give the cleaned bunch a rinse. Place them in a small saucepan with ¾ cup water, bring to the boil, and simmer for ten minutes to pop them all open.
Mash the cranberries with a potato masher in the saucepan. Strain through a sieve, pushing the pulp around with a spoon. When it has mostly drained, rinse the pulp with ½ cup of hot water. Repeat. Scrape off the underside of the sieve.
Return the strained juice to the saucepan and heat to dissolve ¾ cup of sugar in it. Stir as you heat to a simmer.
Make a slurry of the potato starch with 1 Tbsp room-temperature water. Stir the slurry into the cranberries and keep stirring as you heat to thicken.
Grind the 1 teaspoon of sugar in a mortar. Pour your pudding into serving vessels (via a funnel, perhaps) and sprinkle the sugar lightly on top to foil its attempt to skin-over. Chill and serve.
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