In a large mixing bowl, mash potatoes using a potato masher or potato ricer. Add in beaten egg, salt and nutmeg, and fold into potatoes. Add 1/2 cup of flour and fold into potatoes. Continue adding small amounts of flour until the mixture comes together to form a smooth dough. Be careful not to overwork the dough. Set aside and let rest for 30 minutes.
Bring a large saucepan of lightly salted water to a simmer.
Divide the dough into 8 balls. Place a sugar cube into the center of the apricot, where the pit had been. Place the apricot into the center of the ball and press the dough up the sides of the fruit until the apricot is fully incased in the dough. Pinch the dough together to seal. Repeat with all of the apricots and balls of dough.
Carefully drop the dumplings into the simmering water. Cook the dumplings for 2 to 4 minutes after they rise to the top, about 12 to 14 minutes. Remove the dumpling from the water with a slotted spoon and allow to drain. Repeat until all dumplings are cooked through.
Bring a large pot of lightly salted water to a strong simmer. Carefully drop in the dumplings, and lower the heat so that it barely simmers. Cook the dumplings for about 10-16 minutes (depending on the size of your dumplings). I like to cook them for a couple of minutes after they rise to the surface. Drain the dumplings and set aside.
In a medium bowl, combine the sugar and cinnamon. In a medium skillet, melt butter. When the butter begins to foam, add the bread crumbs and sauté until golden brown. Add toasted breadcrumbs to cinnamon sugar mixture. Stir to combine. Roll the cooked dumplings in the sugar mixture until evenly coated.
Serves 2 to 4. To serve, melt 2 tablespoons of butter and allow to lightly brown. Drizzle browned butter over the dumplings.