Gather your ingredients and preheat the oven to 350 degrees F (175 degrees C).
Melt butter in a skillet over medium-low heat. Add onion, celery, and carrots and cook, stirring occasionally, until tender, about 15 minutes.
Stir in peas, flour, parsley, and thyme and cook, stirring constantly, until the flour coats the vegetables and begins to fry, about 5 minutes.
Whisk in chicken broth and half-and-half, and cook until the sauce is thick and bubbling. Stir in cooked chicken, and season with salt and pepper.
Transfer chicken, vegetables, and sauce into a 7x11-inch baking dish. Arrange biscuits on top of the filling.
Beat egg yolk with water in a small bowl. Brush mixture over the biscuits.
Bake in the preheated oven until the biscuits are golden brown and the pie filling is bubbly, 20 to 25 minutes.
Remove from the oven and let rest for 10 minutes before serving.