Rinse the duck carcasses under cold running water, then dry them. Preheat the oven to 190°C (375°F) and roast the carcasses on a baking tray lined with parchment paper for about 30 minutes.

Place the roasted carcasses in a pot and add the vegetables and aromatics: carrot, onion, leek, celery root, celery stalks, parsley, garlic, fresh thyme, black peppercorns, allspice, and bay leaf.

Cover everything with cold, clean water. Then simmer gently over low heat for 1.5 hours, skimming off any foam occasionally. Towards the end of cooking, season with salt and cook for another 10 minutes.

Cut the carrot and celery stalks into thin sticks. Sauté them briefly in duck fat, just enough to keep their crunchiness.

Fill a pot with water, add vinegar, and season with salt. Bring to a boil. Crack the eggs into the water and poach them on low heat for 2 minutes.

Carefully remove the poached eggs.

To serve, ladle the soup into bowls, add the sautéed vegetables, and gently place the poached eggs on top. Pour over the rich, clear, flavorful roasted duck broth. Garnish with fresh parsley or cilantro leaves to add a touch of brightness and a slight kick to the soup.