Place the potatoes in a large pan of cold salted water and bring to the boil. Cook until tender, then drain well. Allow them to steam dry before peeling and mashing while still warm for a smoother dough.
Transfer the mashed potatoes to a large bowl. Add the egg, salt and semolina, then gradually mix in the flour. Work gently with your hands until a soft but not sticky dough forms. Avoid overworking to keep the dumplings light.
Wash and halve the plums, removing the stones. Mix the caster sugar with cinnamon, then place a small spoonful inside each plum before closing them back together. This step builds the classic sweet centre.
Lightly flour your work surface and roll the dough to about ½ cm thickness. Cut into even squares, roughly 7 to 8 cm each, ensuring they are large enough to wrap around the plums.
Place a filled plum in the centre of each square. Carefully bring the edges together, sealing tightly and rolling gently into a smooth ball. Ensure there are no gaps to prevent splitting during cooking.
Bring a large pot of salted water to a gentle boil. Lower the dumplings in batches and cook until they float to the surface, then continue for another 3 to 4 minutes. Remove with a slotted spoon and set aside.
In a wide pan, melt the butter over medium heat. Add the breadcrumbs and stir continuously until golden and fragrant. Sprinkle in the sugar and mix well, ensuring an even coating base.
Add the cooked dumplings to the breadcrumb mixture and roll them gently until fully coated. Work carefully to keep their shape intact.
Allow the dumplings to sit for a few minutes so the coating settles and the flavours deepen. This brief pause improves both texture and taste.
Serve warm, dusted lightly with icing sugar if desired. Szilvás gombóc pairs beautifully with a spoon of sour cream or a drizzle of melted butter for added richness.