Preheat your oven to 400°F (200°C).
Heat butter in an oven-safe skillet over medium heat. Add diced chicken and cook until lightly browned and cooked through. Remove and set aside.
In the same skillet, add onion, garlic, carrots, and celery. Sauté for 5–7 minutes until slightly tender.
Stir in the flour and cook for 1 minute to form a roux.
Slowly pour in the chicken broth while whisking to prevent lumps. Add milk and stir until thickened, about 3–5 minutes.
Return chicken to the skillet. Add peas, thyme, salt, and pepper. Simmer for 3–4 more minutes.
If using parsley, stir it in now. Remove skillet from heat.
Place puff pastry over the filling and tuck edges inside the skillet. Or, place biscuits on top.
Brush pastry or biscuits with beaten egg if using.
Bake for 20–25 minutes or until golden brown and bubbling around the edges.
Let rest for 5 minutes before serving.