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Place a rack in the center of the oven and preheat the oven to 400 degrees F.
Place the potatoes in a medium saucepan. Cover with water by 2 inches, then add the vinegar and 1 1/2 teaspoons salt. Bring to a boil, lower the heat, then simmer for 5 minutes. The potatoes should be fork tender but not total mush. Drain and set aside.
Meanwhile, heat a large (12-inch), cast iron or similar oven-safe skillet over medium heat. If using chicken or turkey sausage, add 2 teaspoons olive oil (if using pork sausage, you may not need the oil). Add the sausage, and sauté for 6 to 8 minutes, breaking the meat into small pieces, until lightly browned and cooked through. If the skillet looks dry at any point, drizzle in additional oil to keep the pan from drying out. With a slotted spoon, transfer to a plate.
Add 1 tablespoon oil and bell peppers to the pan. Increase the heat to medium-high and let cook for 4 minutes, until they are beginning to soften.
Add the drained sweet potatoes to the pan. Cook, tossing occasionally with a spatula, for 6 to 8 minutes, until the sweet potatoes are lightly browned. If the pan becomes dry at any point, drizzle in more oil as needed to prevent sticking. Return the sausage to the pan. Add half of the green onion, sriracha, black pepper, and the remaining 1/2 teaspoon salt. Stir and let cook 2 minutes.
Create 4 wells in the hash where you will crack each egg. Working one egg at a time, crack an egg into a small bowl, then carefully pour it into one of the wells. Quickly repeat with the remaining eggs.
Transfer the skillet to the oven and bake until the eggs are done to your liking, about 10 to 15 minutes for lightly runny yolks. Sprinkle with the remaining green onions and any desired toppings. Scoop generously onto plates, ensuring each serving has an egg. Enjoy immediately.