Rinse and pat the meat dry with paper towels. If using a whole leg, season with salt and pepper and rub with olive oil, then heat 1/3 cup of the olive oil in a large flameproof baking pan over high heat. Add the meat and sear in the oil, turning to brown on all sides. If using stewing pieces, season with salt and pepper, then heat 1/3 cup of the olive oil in a Dutch oven or large stewing pot and brown the meat over high heat. Remove from the pan or pot. Preheat the oven to 375 degrees.
Meanwhile, heat 1/3 cup of the remaining olive oil in a large, wide pot over medium heat, add the scallion greens, reduce heat to low, and cook, stirring, until wilted, 7 to 8 minutes. Season with salt and pepper and add the paprika. Cook, stirring, until very soft and well blended with the paprika, another 10 to 12 minutes or so. Remove from the heat and mix in the mint.
Spread the scallions over the bottom of a large roasting pan. Place the meat (leg or stewing pieces) on top and place in the oven. Adjust the seasoning with salt and pepper, add the water, and roast, covered with aluminum foil, until the meat is very tender, about 1 1/2 hours. About 15 minutes before the meat is ready, remove the foil, beat the eggs and pour over the meat. Continue baking until the eggs form a crust. Remove from the oven, let cool slightly, and serve.