In a medium size bowl, mix the sour cream, ketchup and relish to make a dressing. Season the Russian dressing with salt and pepper.
In a heavy skillet, heat 1 tablespoon oil over medium-high. Melt the butter in the oil, then add the potatoes and cook until browned, 8-10 minutes. Sprinkle with the caraway seeds; season. Add the onion; stir for 2-3 minutes. Add the turkey and cook until lightly browned at the edges, 2-3 minutes more. Stir in the sauerkraut; cook to heat through. Sprinkle with the Gruyère , then tent with foil to melt the cheese.
In a large nonstick skillet, heat a little oil over medium. Add the eggs (1 or 2 per person) and cook, flipping once, to desired doneness, 3-7 minutes.
Divide the hash among plates. Drizzle with the Russian dressing; top with the eggs.