Pick all the meat off and place it in a separate container in the refrigerator.
Next, I place the carcass and bones into a large stockpot and cover with 1/2 water and 1/2 chicken broth, just enough to not quite cover the bird. Your house will smell amazing, just a warning!
Bring this to a boil, then reduce heat, cover the pot and allow to simmer for around 1 and 1/2 hours.
After the stock has cooked, drain broth through a colander lined with cheesecloth and discard bones. Return stock to a large stockpot. Add enough canned chicken broth if needed to have 10 cups of stock.
Sauté garlic and onions in olive oil in a saucepan until onions are transparent, about 5 minutes on medium heat. Add cooked onions and garlic to stock.
Add shredded chicken, carrots, bay leaves, basil, thyme, greek seasoning, parsley and 3 tablespoons of fresh lemon juice to stock.
Simmer 15 minutes
Add rice, cook until tender; about 20 minutes.
To make the avgolemono, place the eggs in a medium bowl and whisk until lightened in color and frothy, about 2 minutes. While whisking, slowly pour in the lemon juice. While still whisking, temper in the eggs by slowly drizzling the reserved 2 cups of warm stock into the egg-lemon mixture. This warms the eggs just enough so that they do not curdle when added to the hot soup.
Add the avgolemono back into the pot with the chicken and rice and stir to combine. Cook until the soup thickens slightly, 3 to 5 minutes, but do not let it come to a boil.
Discard bay leaves, add peas, cook 3 minutes and serve.