In a large nonstick frying pan, heat the oil over moderately high heat. Add the onion and thyme and cook, stirring, until softened and beginning to brown, 3 to 5 minutes. Add the sweet potatoes, turkey, salt and some pepper and cook, stirring occasionally, until the potatoes are beginning to brown, about 3 minutes longer. Add the broth, scraping up any brown bits that may have formed on bottom of pan. Keep warm while poaching the eggs.
Fill a saute pan with 1 1/2 inches of water and the vinegar and bring to a simmer. Break 1 egg into a small bowl and then carefully pour it into the water. Repeat with the remaining eggs spacing them so they do not touch. Poach at a gentle simmer until the whites are firm but the yolks are still runny, 3 to 4 minutes. With a slotted spoon, transfer the eggs to paper towels and season with salt and pepper.
Divide the hash between 4 plates and top each serving with a poached egg. Serve with hot sauce.