Pick the carcass. Before anything else, pick off any remaining meat from the bones and set it aside. Get into the joints and the back — there's usually more there than you think. You don't need to get it all perfectly, just the pieces that come off easily. The rest will fall off during simmering and you can pick those out after straining. Then place carcass in a large stockpot with the olive oil and cook for 2-3 minutes per side to brown it and build a deeper flavor.
Make the stock. Cover with 12 cups of cold water. Add the first set of vegetables (½ onion, 2 celery ribs, 2 carrots), 1 tsp salt, and the apple cider vinegar if using. Bring to a gentle boil over medium heat, then reduce to a low simmer. You want the surface barely moving — a full rolling boil makes the broth cloudy and slightly bitter. Simmer for 2 hours, skimming any foam or grey impurities off the surface every 30 minutes or so with a slotted spoon.
Strain the broth. After 2 hours the broth should be golden and smell incredible. Set a fine mesh strainer or cheesecloth over a large bowl or second pot and pour the broth through it. Discard all the solids — the carcass, the first set of vegetables — they've done their job and have nothing left to give. Pick any remaining meat off the carcass before discarding the bones.
Build the soup. Return the strained broth to the pot over medium heat. Add the second set of vegetables (½ onion, 4 celery ribs, 4 carrots, all diced), the bay leaf, basil, and oregano. Simmer for 20 minutes until the vegetables are tender but not mushy. Add the shredded chicken and taste for salt and pepper — the broth will likely need more salt than you think.
Cook the pasta separately. In a separate pot, cook the pasta according to package directions until just al dente. Drain and add to the finished soup. Cooking pasta separately prevents it from absorbing all the broth as the soup sits — especially important if you're making this for leftovers.
Serve. Ladle into bowls and top with fresh parsley. Serve with crusty bread or gluten-free bread on the side. This soup is better the next day.