If using Italian prune plums, halve them lengthwise and remove the pits. Fill each cavity with 1/2 teaspoon of the sugar and re-form the plums by pressing the halves together. If using larger plums, halve them crosswise, separate the halves neatly, and remove the pits with a small melon baller. Cut them into 1/2-inch dice. In a small sugar bowl, toss the diced plums with 1/4 cup of the sugar.
While you prepare the gnocchi dough, bring the water to a boil in a large pot over high heat.
Flour your hands, the dough, and the work surface. Roll the dough under your palms to form a cylinder 2 inches in diameter and slice it evenly into 24 rounds. Flatten each round in the palm of one hand to a circle about 5 inches in diameter. Place an Italian plum or about 2 tablespoons of the sweetened diced plums (or 2 tablespoons of the jam) in the center of each dough circle. Carefully gather the dough up around the fruit (or jam), enclosing it completely without tearing the dough. Pat the dough between your hands to seal and even the gnocchi. Flour the dough and your hands lightly as you work and place the gnocchi on a lightly floured baking sheet as you form them.
Add as many of the gnocchi as will fit without touching (about half) a few at a time to the boiling water, stirring gently with a wooden spoon to prevent sticking. Cook until the dough is tender and the plums are very soft, about 8 minutes after they rise to the surface.
Meanwhile, in a heavy, medium-size skillet, melt the butter over medium heat. Add the bread crumbs and toast, stirring almost constantly, until golden brown, about 7 minutes. Remove from the heat, add the remaining 1/2 cup sugar and the cinnamon, and blend thoroughly. Transfer to a large baking dish.
Remove the cooked gnocchi with a slotted spoon, draining them well, and repeat with the remaining gnocchi. Roll the cooked and drained gnocchi while still warm in the bread-crumb mixture until all are well coated. Arrange on a serving plate and sprinkle with any bread crumbs remaining in the baking dish. Serve warm.
Place the potatoes in a large pot with enough cold water to cover. Bring the water to a boil and cook, partially covered, until the potatoes are easily pierced with a skewer but the skins are not split, about 35 minutes. (Alternatively, the potatoes can be baked in a preheated 400-degree oven until tender, about 40 minutes.)
Drain the potatoes and let them stand just until cool enough to handle. (The hotter the potatoes are when they are peeled and riced, the lighter the gnocchi will be.) Working quickly and protecting the hand that holds the potatoes with a folded kitchen towel or oven mitt, scrape the skin from the potato with a paring knife. Press the peeled potatoes through a potato ricer. Alternatively, the potatoes can be passed through a food mill fitted with the fine disc, but a ricer makes fluffier potatoes and therefore lighter gnocci. Spread the riced potatoes into a thin, even layer on the work surface, without pressing them or compacting them. Let them cool completely.
In a small bowl, beat the egg, salt, and nutmeg together. Gather the cold potatoes into a mound and form a well in the center. Pour the egg mixture into the well. Knead the potato and egg mixtures together with both hands, gradually adding enough of the flour, about 1 1/2 cups, to form a smooth but slightly sticky dough. It should take no longer than 3 minutes to work the flour into the potato mixture; remember, the longer the dough is kneaded, the more flour it will require and the heavier it will become. As you knead the dough, it will stick to your hands and to the work surface: Repeatedly rub this rough dough from your hands and scrape it with a knife or dough scraper from the work surface back into the dough as you knead. Dough is ready to be filled.