Boil 2 pounds potatoes in their skins until tender, ideally a day ahead. Let them cool completely before making the dough.
Bring a large pot of lightly salted water to a boil.
Peel the cold potatoes and grate them finely, or press them through a potato ricer. Add 1 egg.
Mix 2 ½ cups all-purpose flour, 1 Tablespoon potato starch, ¼ teaspoon baking powder, and 1 teaspoon salt in a separate bowl. Add the mixture to the potatoes and work quickly, with dry hands, into a smooth dough.
On a floured surface, roll the dough into a log and cut it into equal pieces. Flatten each piece into a disc, put the two halves of a plum back together in the middle, close the dough around the fruit, seal it, and roll the dumpling between your palms.
Lower the dumplings into the water in batches and cook for 8-12 minutes, depending on size. Keep the water at a gentle bubble, not a rolling boil.
Lift one dumpling out, cut it open, and check that the dough next to the plum is cooked through.
Serve hot, sprinkled with ground poppy seeds and powdered sugar and drizzled with melted butter.
Notes
Cold potatoes: the potatoes must be completely cold before you make the dough, or it turns loose and sticky; boiling them a day ahead works best.
Dry hands: work fast and keep your hands dry; wet hands make the dough fall apart.
Sour plums: tuck a sugar cube into the dumpling in place of the pit.
Store: fridge up to 3 days; freeze without the topping, first on a tray, then in a bag.
Serving suggestion:As a main course, count on about three large dumplings per person, with roughly two tablespoons of the poppy and sugar mixture and two of melted butter each; anyone who likes it sweeter and richer can simply add more. For a dessert-sized taste, one dumpling is plenty.