In a large soup pot, heat oil over medium-high heat. Add onion, carrots and celery and cook and stir for 3-4 minutes, until onion begins to brown

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 (a simmer), cover, and cook 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.