Preheat oven to 375 degrees.

1. Melt the butter in a large skillet.  Add the onion and celery and saute until the celery and onions are tender.  Add the garlic and cook for another minute.  Remove from heat and set aside.  (You can use a separate skillet for the next step or transfer the onions and celery to a bowl and use the same skillet).
2. Add 1/2 cup of the broth and the flour to a large skillet.  Whisk over medium-high heat until the mixture thickens.  (This is called making a roux).  Add the remaining broth,  Whisk until the mixture is slightly thickened and remove from the heat.  Add the half & half, salt, pepper, and Thyme, and mix to combine.  Add the peas & carrots, and chicken and mix everything together, making sure everything is lightly coated with the gravy.  Transfer the mixture to a lightly buttered 10-inch iron skillet.
3. Make the pie crust.  (If you’re using a homemade pie crust, such as the one above, you will only have to use 1/2 of it, enough for a single crust).
4. Roll out the pie dough and place it over the pie filling.  Trim the edges to 1/2 inch past the edge of the skillet.  Fold under and flute or crimp as desired. Cut 4 to 5 slits in the crust for venting.  Brush the top of the crust with a simple egg wash if desired.  I used one egg and 1 tablespoon of water mixed together and brushed it over the top.
5.   Bake in the preheated oven until the top is golden brown, about 45 minutes, depending on your oven.
6.   Allow pie to rest for 5 minutes before cutting and serving.
7. Refrigerate leftovers.  Pie will keep for up to 5 days, refrigerated.