Prepare the ingredients and preheat the oven to 350°F (175°C).
Melt butter in a skillet over medium-low heat. Add the onion, celery, and carrots, cooking and stirring occasionally, until the vegetables are tender, about 15 minutes. This step allows the vegetables to soften and develop a sweet, aromatic flavor that forms the base for the filling.
Stir in the peas, flour, parsley, and thyme, and cook, stirring constantly, until the flour coats the vegetables and starts to lightly brown, about 5 minutes. This helps create a roux-like mixture, which thickens the filling and deepens the flavors, setting the foundation for a creamy consistency in the dish.
Whisk in the chicken broth and half-and-half, then cook until the sauce thickens and starts to bubble. Stir in the cooked chicken, and season with salt and pepper to taste. This step creates a creamy, flavorful sauce that binds the ingredients together, while the chicken adds richness and texture to the filling.
Transfer the chicken, vegetables, and sauce into a 7x11-inch baking dish. Arrange the biscuits evenly on top of the filling. This step ensures that the filling is evenly distributed, while the biscuits will bake to a golden, fluffy finish, creating a delicious, comforting crust over the rich, savory filling.
Whisk the egg yolk with water in a small bowl. Brush the mixture over the biscuits. This will give the biscuits a golden, glossy finish as they bake, enhancing their appearance and adding a slight richness to the texture.
Bake in the preheated oven for 20 to 25 minutes, or until the biscuits are golden brown and the pie filling is bubbling. This ensures the biscuits are fully cooked, with a crisp, golden top, while the filling becomes hot and bubbly, creating a comforting, hearty dish.
Remove from the oven and let the dish rest for 10 minutes before serving. This allows the filling to set, making it easier to serve, while the biscuits cool slightly to the perfect texture.