Place baseball-sized portions of grated potatoes in a thin linen towel or cheesecloth. Wring out as much water as possible, transferring each strained batch into a large bowl.
Alongside the strained potatoes, add starch and salt. Using clean hands, knead to form a just barely sticky dough. If potato mixture feels very wet, add a little more starch.
Heat a large pot of salted water to boiling over high heat. Place a colander in a large bowl; set aside.
Spread a layer of potato dough on a small cutting board, about 1/2-inch thick. Dip a soup spoon into the boiling water and, working in batches, scrape small dumplings directly from the cutting board into the boiling water, wetting the spoon with boiling water in between. Allow dumplings to boil until they float to the top, about 3 minutes. Transfer dumplings to the prepared colander using a skimmer. Repeat, boiling in batches, until the dumplings have all been boiled.
Melt 2 tablespoons butter in a large skillet over medium-low heat, add 2 onions, and cook, stirring occasionally, until caramelized, about 30 minutes.
Line a large plate with paper towels. Cook bacon in a small skillet over medium heat until crispy, about 10 minutes. Transfer bacon to the prepared plate with a slotted spoon.
Melt remaining 2 tablespoons butter in a large nonstick skillet over medium-high heat, then add drained potato dumplings. Cook, stirring occasionally, until light golden brown, about 3 minutes. Crack eggs directly into the skillet and stir until eggs are scrambled and cooked through, 1 to 2 minutes. Season with salt.
Serve eggs and potato dumplings topped with caramelized onions and bacon.