Heat oil over medium flame in a Dutch oven or heavy soup pot until shimmering. Add onion, celery and carrots, and stir to coat with oil. Cook vegetables until starting to soften, about 5 minutes, stirring frequently to avoid browning or burning.
Clear a space in the middle, add garlic, oregano and thyme, and cook, stirring constantly, about 45 seconds. Add chicken stock, season with salt and pepper, and bring to a boil. Reduce to a simmer, stir in rice and cook for about 20 minutes, stirring occasionally.
Add chicken and cook until chicken is warmed through, about 5 minutes.
Meanwhile, make the avgolemono sauce. Whisk the egg yolks and lemon juice together in a deep bowl or measuring cup. Continuing to whisk, slowly ladle in 2 ladlesful of broth. This tempers the egg lemon mixture and keeps it from cooking into solids.
Turn the heat off under the Dutch oven and whisk the avgolemono sauce into the soup. Taste and adjust seasoning with salt.
Ladle into shallow soup bowls and serve.