In a large pot bring water to boil and add in the carrots and potatoes. Cook until just slightly softened and then drain. Set the cooked potatoes and carrots aside.
In a large pot melt the butter over medium-high heat. Add in the onions, celery, and broccoli.
Cook until softened about 4-5 minutes
Sprinkle the cooked onions, celery and broccoli with flour and cook one minute.
Slowly add the broth, milk, salt, and pepper.
Stir well to combine.
Add in the cooked chicken, potatoes, thawed peas and carrots.
Cook over medium heat for about 10-15 minutes.
While the soup is cooking, roll out the single pie crust and cut into strips.
Place on a lined cookie sheet and bake at 450 degrees for 6-8 minutes or until golden brown.
Once soup is done cooking. Serve hot with pie strips and garnish with dairy-free cheese.