Melt 3 tablespoons of butter in a frypan over medium-low heat, cook and stir the bread crumbs until golden brown and fragrant; about 2 minutes. Set the crumbs aside.
Place potatoes in a pot of water over medium heat and boil until tender, 20 to 30 minutes. Drain the potatoes and allow to cool for several minutes to dry out; then mash in a bowl. Place 1 tablespoon of butter into the potatoes and allow to melt, then mix in the flour until thoroughly combined. Mix in egg and salt. Turn the mixture out onto a generously floured work surface and knead until the dough is soft and no longer sticky, about 10 minutes.
Divide the dough into quarters and subdivide each quarter into fourths to make 16 portions. Roll each portion into a ball and roll the ball out on a floured work surface until it forms a circle about 10cm in diameter. Place a pitted plum into the centre of the dough circle and spoon a teaspoon of sugar into the plum. Roll and pinch the dough around the plum to seal. Repeat with the remaining dough to make 16 dumplings.
Bring a pot of lightly salted water to a boil over medium heat and drop the dumplings into the boiling water. Stir gently to loosen any dumplings that stick to the bottom. Allow the dumplings to rise to the top then boil for 5 more minutes. Gently remove dumplings with a slotted spoon and roll in the toasted bread crumbs to serve. Any remaining dough can be rolled into plain dumplings, boiled, and rolled in crumbs.
Instead of spooning a teaspoon of sugar into each plum, it's easy to push a sugar cube into the plum.