Grate potatoes on the largest hole on a box grater into a medium bowl. Cover with cold water and drain; repeat once or twice, until water runs nearly clear. Taking small handfuls at a time, squeeze all the water from the potatoes and place in a dry medium bowl. Grate onion on the largest hole on a box grater and toss with potatoes. Season with salt and pepper and let sit while you cook the sausage.
Cook sausage in a large cast-iron skillet over medium heat until totally crispy, about 3 minutes per side. Remove from pan and put on a plate lined with paper towels and set aside.
Taking small handfuls at a time, squeeze all water from the potato-onion mixture. Heat 2 tablespoons vegetable oil in that same skillet over medium heat, add 2 tablespoon butter and swirl together.
Add potatoes and onions and spread into a thin, even layer. Cook, without moving, until the potatoes are nearly cooked through and browned on one side, about 5 to 8 minutes. Using a spatula, flip sections of the potatoes to crisp on the other side. Add 2 tablespoons vegetable oil and remaining 2 tablespoons butter and cook, without moving, until potatoes are browned on the other side, about another 5 minutes.
Transfer potatoes to a plate and set aside while you fry the eggs. Meanwhile, toast the bread.
Increase heat to medium high and add 1 tablespoon vegetable oil. Crack 4 eggs at a time into the skillet and season with salt and pepper. Cook, rotating skillet occasionally until the whites are cooked through and the yolks are still bright yellow and runny, 2 to 3 minutes. Repeat with the other 4 eggs.
Place on 4 plates with hash browns, sausages and toast. Serve with hot sauce or ketchup as desired.