In a large pot, heat oil over medium-high heat. Add onion, carrots, and celery and cook for 3-4 minutes, until onion is translucent.
Add garlic, parsley, and thyme, and cook 1 minute.
Add broth, chicken, rice, salt, and pepper. Stir and bring to a boil over medium-high heat.
Reduce heat to medium-low and cover. Simmer for 30 minutes or until vegetables and rice are tender and chicken has reached 165℉.
Remove chicken from pot and shred. Add back to the pot with evaporated milk or cream if using.
