In a large pot, heat oil over medium-high heat. Add onion, carrots, and celery and cook and stir for 3-4 minutes, until onion begins to turn golden.
Add garlic, parsley and thyme and cook 1 minute.
Add salt and pepper, broth, chicken. Add rice. Stir and bring to a boil over medium-high heat.
Reduce heat to medium-low and cover. Simmer for for 30 minutes, stirring every 10 minutes, or until vegetables and rice are tender.
Remove chicken from pot and shred. Add back to the pot with evaporated milk or cream if using.
