Place potatoes in a small pot and cover with water. Bring to a boil and cook until fork tender, about 6 to 8 minutes. Drain and set aside.
Melt butter over medium heat in a Dutch oven.Add onion, carrot, and celery and cook for 5 to 6 minutes, stirring frequently.
Add garlic and flour and cook and stir for 1 minute.
Gradually whisk in the broth and half-and-half. Add salt, pepper, and poultry seasoning. Bring to a simmer. Simmer for 2 to 3 minutes to thicken.
Stir in chicken, peas, and potatoes. Add cheese and stir in until melted. Remove from heat. Check for seasoning and add more salt if desired.
Unroll one piecrust and fit it into a 9-inch deep dish pie pan. Place filling in crust.
Unroll second pie crust and lay it on top of the pot pie.Crimp the edges of the two crusts together around the edges. Brush with egg.
Take a knife and cut 4 slits in the pie crust near the center.Place pie plate on a rimmed baking sheet and bake for 30 to 35 minutes, or until golden brown.