Boil whole potatoes with their skins on in a big pot of water seasoned with 3 tsp of the salt until just tender, about 45 minutes.
While the potatoes are cooking, prepare the plums. Cut each plum in half, remove the pit, sprinkle about ¼ teaspoon of sugar in the center where the pit was, and put back together. If using large plums, chop into small pieces and toss with sugar.
Peel cooked potatoes and pass through a food mill or ricer onto a large wooden cutting board. Spread out to cool.
Gather up the cooled potatoes into a mound and make a well in the center. Add the egg and 1 tsp salt and begin mixing with a fork, slowly incorporating the potatoes into the egg. Once the egg is absorbed, begin adding the flour ½ cup at a time, working it in gently after each addition. Continue adding flour until the dough is no longer sticky.
Form dough into a ball, and then roll out into a sheet about ½" thick. Cut into squares about 2" square.
Place one plum in the center of each square and then carefully wrap the dough around the plum, forming a large ball. Make sure that all of the seams are sealed well.
Bring a large pot of salted water to a boil. Carefully add the gnocchi one at a time using a spider or strainer. Cook for about ten minutes after they rise to the surface, till the gnocchi are somewhat firm but still tender. They will still be quite delicate though, so take care not to smoosh them, but stir often to prevent sticking.
While the gnocchi cook, melt the butter in a large saucepan. Add the breadcrumbs and cook till golden brown. If sauce seems too thick, add more butter.
When gnocchis are cooked, carefully drop them into the pan with the breadcrumbs and cook over high heat for about 4 or 5 minutes, carefully flipping them over halfway through. You want to get just a little bit of a crust on the top and bottom.
Place 3 gnocchis in each bowl and drizzle lots of butter and breadcrumbs on top. Finish with a sprinkling of cinnamon and sugar.