Heat 2 teaspoons oil in a pot over med-high heat; season the chicken with salt and add it to the pot; cook/stir occasionally, until just cooked through, about 5 minutes; transfer the chicken to a dish; set aside.
Add the remaining 2 teaspoons oil to the pot; add in onions, celery, carrot, and thyme; cook/stir over med-high heat until the vegetables are tender, about 5 minutes.
Add in 5 cups broth and bring to a boil; add in orzo and let simmer until tender, about 8 minutes.
Decrease heat to low to keep the soup hot, but not boiling.
Warm the remaining 1 cup broth in a small pan until it is hot, but not boiling.
In a 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 in cooked chicken; season to taste with salt and pepper; serve.