Place the turkey carcass in a large stockpot. Add 10 cups of water and bring to a boil over medium-high heat.
Once boiling, reduce the heat to low and skim off any foam that rises to the top.
In a separate pan, heat olive oil over medium heat. Add the chopped onion, garlic, carrots, and celery. Sauté for about 5-7 minutes until softened.
Add the sautéed vegetables to the stockpot with the turkey carcass. Stir in the potato, green beans, thyme, rosemary, bay leaf, and salt and pepper.
Cover the pot and let the soup simmer on low heat for about 1.5 - 2 hours, allowing the flavors to meld and the turkey to become tender.
After cooking, carefully remove the carcass from the pot. Let it cool slightly, and then shred any remaining meat from the bones with a fork.
Return the shredded turkey meat to the pot, and if you are adding pasta or rice, include it at this point and cook according to the package instructions.
Taste and adjust seasonings if necessary. Remove the bay leaf before serving.