Boil potatoes and carrots in a small saucepan for 5–8 minutes until tender, then drain and set aside.
Melt butter in a large skillet, add onions, and sauté 2–3 minutes. Stir in salt, pepper, thyme, and poultry seasoning.
Sprinkle flour over onions, stirring for 1 minute to create a roux.
Slowly whisk in chicken broth and milk until smooth and slightly thickened.
Add cooked vegetables, shredded chicken, diced ham, and peas. Mix well and remove from heat.
Line a pie plate with one crust, pour in filling, and top with second crust. Seal and crimp edges; cut slits for steam.
Brush top crust with beaten egg and water for golden color.
Place pie on baking sheet and bake at 425°F (220°C) for 30 minutes or until crust is golden and filling bubbles.
Let rest 10 minutes before slicing and serving.
