Boil 1 pound potatoes in their skins until they are soft, which usually takes about 20 minutes depending on their size. Check for doneness by piercing a potato with a fork; if it's soft, it's cooked. Drain the water and let the potatoes cool completely.
Peel the cooled potatoes and finely grate them using a box grater. Add 1 egg, 2 cups all-purpose flour, and ¼ teaspoon salt to the grated potatoes.
Work the mixture into a compact dough. Start by roughly mixing all the ingredients in a bowl using a fork. Then transfer the mixture onto a floured work surface and knead until the dough is no longer sticky.
If the dough is too sticky, dust it with a tablespoon of flour and work it in. Continue adding flour a little at a time until the dough holds together nicely and no longer sticks to your hands.
Divide the dough into three to four pieces. Roll each piece into a cylinder. Cut equal-sized pieces from each cylinder and roll them into Šiška shapes, about the size of your index finger.
Meanwhile, take a wide pot, pour water into it, add 1 teaspoon salt, and bring it to a boil on the stove. Once it reaches a boil, reduce the heat so that the water simmers.
Carefully drop the Šišky into the simmering water. They will stick to the bottom initially; gently peel them off with a fork. Once the Šišky float to the surface, cook for an additional two minutes. Transfer them to a shallow bowl using a colander or a slotted spoon. The entire cooking process takes about 10 minutes.
Fry the breadcrumbs with sugar: Melt 2 Tablespoons unsalted butter in a frying pan over medium heat. Add 1 ¼ cups breadcrumbs and stir continuously. If the breadcrumbs do not turn golden within five minutes, slightly increase the heat. Once golden, remove the pan from the heat and add ½ cup granulated sugar to the breadcrumbs, stirring well to combine. The mixture will be crunchy.
Serve the Šišky warm, sprinkled with sweet breadcrumbs. For an extra touch, drizzle them with a little melted butter before serving.