Preheat your oven to 375 degrees. Spray a 2 quart casserole dish with cooking spray.
Melt the butter in a large skillet or a dutch oven. Add the carrots, onion, celery, and garlic. Season with salt and pepper and sauté until the vegetables are soft, about 5 to 7 minutes.
Whisk in the flour. (It will look very gummy, but you are on the right track.) Then slowly whisk in the chicken stock, scraping up any browned bits. Whisk in the milk, thyme, and add in two bay leaves. Bring to a simmer.
Add in the chicken and cook for 10 minutes or until the chicken is cooked through. Discard the bay leaves and stir in frozen peas. Pour into the prepared casserole dish.
Top the casserole dish with the puff pastry. Then brush the puff pastry wth the egg wash. (Read here for tips about doing a lattice top.)
Bake for 25 to 35 minutes or until the puff pastry is golden brown. Allow to cool for 10 minutes before dishing up.