Break the chicken carcass into several pieces so it fits compactly in a 5- to 6-quart heavy-bottomed pot or Dutch oven, breaking apart smaller bones and joints as needed.
Heat the olive oil in the pot over medium-low heat. Add the chicken carcass, neck and giblets if using, onion, carrots, celery, garlic, bay leaf, peppercorns, cloves, smoked paprika, cayenne and salt. Cook gently for 3–5 minutes, stirring occasionally, until the vegetables and aromatics become fragrant and begin to lightly brown.
Add the apple cider vinegar and 10 cups of water, or enough to cover the contents of the pot by about 1 inch.
Bring just to a gentle simmer, then reduce the heat to low. Partially cover the pot and maintain a very gentle simmer for at least 4 hours and preferably 7–8 hours. Check the liquid level occasionally and add a small amount of water if needed to keep the bones mostly submerged, taking care not to dilute the broth unnecessarily.
Remove the larger bones and vegetables, then carefully pour the broth through a large strainer into a heat-safe bowl or container.
Taste and add additional salt if desired. Pour the broth through a fine-mesh strainer as you divide it among smaller jars or storage containers. Allow it to cool before refrigerating, making sure it isn't left at room temperature for more than 2 hours. Refrigerate and use within 3–4 days or freeze for longer storage.