Make the stock by adding the chicken, carrots, celery, onion, 2 teaspoons salt, oregano, and thyme to a large soup pot. Add enough cold water to cover the ingredients. Bring to a boil over medium heat.
Once boiling, reduce the heat to low, cover with a lid, and cook the stock until the meat falls off the bone, about 60 minutes.
Turn off the heat and strain the broth, discarding the carrot, celery, onion, and thyme stems and reserving the chicken for later. Reserve 1 cup of broth for later, then return the rest of the broth to the pot.
Add the rice to a bowl, cover it with water, and soak for 10 minutes.
Meanwhile, shred the chicken and set it aside on a plate.
Bring the chicken broth back to a simmer over medium heat. Add the soaked rice to the broth and cook, covered, for about 7 minutes until the rice is cooked through.
While the rice is cooking, whisk the eggs and lemon juice in a bowl to combine. Gradually whisk in the reserved cup of broth to temper the eggs. The result should be a creamy, frothy liquid.
Stir in the egg mixture into the large pot of broth and add the shredded chicken. Cook over medium heat until the chicken warms through, about 5 minutes.
Taste for seasoning, adding more salt if desired.
Garnish avgolemono with additional thyme, if desired, and enjoy while hot.