Mom's Fabulous Chicken Pot Pie with Biscuit Crust

butter

small onion, chopped

celery ribs, chopped

carrots, chopped

frozen peas

chopped fresh parsley

dried thyme

all-purpose flour

lower-sodium chicken broth

half-and-half cream

salt and ground black pepper

cooked chicken, cut into bite-size pieces

refrigerated flaky-style biscuits (such as Pillsbury Grands®)

egg yolk, beaten

water