Preheat large pot on medium-high heat and add oil. Add chicken breasts, sprinkle with salt and pepper and cook for 3 minutes per side. Transfer onto a plate.
Add butter and mushrooms, saute for 3 minutes, stirring once. Then add garlic, onion and celery. Saute for 5 minutes, stirring occasionally. Add thyme and cook for 30 seconds, stirring a few times. Add flour and stir until combined.
Add chicken broth, stir and bring to a boil. Scrape the bottom of the pot well and add previously cooked chicken, carrots, potatoes, bay leaves, salt and pepper.
Stir, cover and bring to a boil. Reduce heat to low and simmer for 15 minutes. Open the lid and simmer for 5 more minutes.
Remove chicken breasts onto the plate. To the pot, add milk and green peas. Let cook for 5 minutes while you are shredding the chicken with 2 forks.
Return shredded chicken to the pot along with parsley, stir and adjust any seasonings to taste.
Serve hot with biscuits, dinner rolls or butter crackers, or anything bread like.
Notes
Store: Refrigerate in an airtight container for up to 5 days. Reheat desired amount by simmering on low heat while covered.
Freeze: Freeze in an airtight container, leaving some room for expansion, for up to 3 months.