Put the sliced leeks in a large bowl of cold water and let them soak for 10 minutes. Lift them out carefully, making sure to leave the grit at the bottom of the bowl behind, and set aside (there's no need to dry them). Put the stock and carrots in a large saucepan. Bring to a boil, reduce the heat, and cover the pot. Simmer for 5 minutes and then add the leeks and ginger. Continue to cook, covered until the vegetables are just tender, about another 5 minutes. Add the diced turkey, lemon juice, salt, and pepper. Simmer until the turkey is heated through, about 2 minutes. Adjust the seasonings if needed, add the shredded mint, and serve immediately.

Position a rack in the middle of the oven and heat the oven to 425 degrees F. Pour the vegetable oil into a large flameproof roasting pan. Break or chop the turkey carcass into 3 or 4 pieces and put it in the roasting pan, along with the onion, celery, and carrot. Roast for 30 minutes, stirring two or three times to ensure even browning. Transfer the turkey and vegetables to a large stockpot. Pour off and discard any fat from the roasting pan, set the pan over medium heat, and add the brandy. Stir with a wooden spoon, scraping up all the browned bits from the bottom of the pan. When the mixture is bubbling, pour the drippings into the stockpot. Add the ginger, bay leaf, thyme, and peppercorns to the pot. Add about 12 cups cold water (or enough to almost cover the turkey pieces). Bring to a simmer, skim any foam that rises to the top, and then reduce the heat to a very slow simmer. Simmer for 2 hours (if you used more than 12 cups water, you may need to boil it down a bit further for flavor). Strain into a large bowl, cool, and refrigerate overnight. The next day, skim the fat from the top of the stock and then portion it as you like.