For the Chicken and Broth:

• 1 (4 ½-pound) whole chicken, giblets removed
• 2 celery stalks, cut into large pieces
• 1 onion, quartered
• ¼ cup chopped parsley
• 4 sprigs fresh thyme
• 2 bay leaves
• 1 tablespoon salt
• ½ teaspoon ground black pepper

For the Filling:

• ½ cup butter, divided
• 1 cup chopped celery
• 1 cup chopped carrots
• 1 cup chopped onion
• 1 (8-ounce) package sliced fresh mushrooms
• 1 cup frozen peas, thawed
• ⅔ cup all-purpose flour
• 4 cups reserved chicken broth
• ½ cup heavy whipping cream
• 4 cups cooked chicken, chopped
• 1 tablespoon minced fresh thyme
• 1 ½ teaspoons salt
• 1 teaspoon ground black pepper

For the Biscuit Topping:

• 2 (7.5-ounce) bags complete buttermilk biscuit mix (such as Bisquick)
• ⅔ cup shredded Parmesan cheese
• ⅔ cup shredded Cheddar cheese
• 1 cup whole milk