Heat 2 teaspoons of the oil in a soup pot over medium heat. Season the chicken with salt, add it to the pot, and cook, stirring a few times, until just cooked through, about 5 minutes. Transfer the chicken to a dish and set aside.
Add the remaining 2 teaspoons oil to the pot. Add the onion, celery, carrot, and thyme and cook, stirring over medium-high heat until the vegetables are tender, about 5 minutes. Add 5 cups of the broth and bring to a boil. Add the orzo and let simmer until tender, about 8 minutes. Turn the heat down to low to keep the soup hot but not boiling.
Warm the remaining 1 cup of broth in a small saucepan until it is hot but not boiling. In a medium bowl beat the eggs. Gradually whisk the lemon juice into the eggs. Then gradually add the hot broth to the lemon-egg mixture, whisking all the while. Add the mixture to the soup, stirring well until the soup is thickened. Do not let the soup come to a boil. Add the cooked chicken to the soup. Season with salt and pepper and serve.