Make Chicken and Broth first. Combine chicken, celery, onion, parsley, thyme, bay leaves, salt, and pepper in a large pot. Add water to cover and bring to a boil. Reduce heat and simmer 45 minutes. Remove chicken and let cool. Discard skin and remove chicken, shredding it into bite-sized pieces. Strain broth and reserve 4 cups.
In a large skillet, melt 1/4 cup butter over medium-high heat. Add celery, carrots, onion, and mushrooms. Cook for 5 to 6 minutes, until vegetables are tender. Stir in peas. Set aside.
Melt remaining 1/4 cup butter in a Dutch oven. Add flour and cook, stirring constantly, for 2 minutes.
Gradually add chicken broth, whisking constantly to avoid lumps. Add cream and continue to cook for 3 to 4 minutes, whisking constantly.
Add reserved chicken, celery/onion mixture, thyme, salt, and pepper. Check and make sure it is seasoned enough and add more salt if necessary.
Pour into prepared dish.
In a medium bowl, combine biscuit mix, cheeses, and milk. Stir just until combines. Take a spoon and drop large spoonfuls evenly over filling.
Bake 18 to 23 minutes, or until biscuits are golden brown.
Notes
As a shortcut, you can use 4 cups of chopped rotisserie chicken and 4 cups of canned broth for this recipe.