Start by cooking your veggies... the carrots, peas (or broccoli in our case) and potatoes. We just combine everything in a large pot of water and boil until everything is just tender, about 4-6 minutes. Then set them aside.
In a large pot or Dutch oven, melt the butter over medium heat. Once melted, add the onion and cook until translucent, about 3-4 minutes. Add the garlic and cook a minute or so more, until fragrant.
Increase the heat to med-high and sprinkle the flour over the onion. Stir to incorporate and cook while stirring for a minute or two. Then, whisking continually, slowly add in the chicken broth and the cream. Stir in all the seasoning and whatever else you'd like and let it come to a low boil.
Once it reaches a boil, reduce the heat down to med-low and allow to simmer until it has thickened, about 7-10 minutes. Taste to make sure it doesn't need any additional seasoning, then remove it from the heat. Stir in the cheese (if using) until it's melted and incorporated in. Then stir in the chicken and veggies.
Preheat the oven to 375F. Lay the first pie dough disc into the bottom of a 9" pie dish. Pour the filling over the top. Lay the second dough disc over the top and trim any overhanging dough from around the edges. Use a fork or your fingers to crimp the edges and seal it.
Use a small, sharp knife to cut a few slits across the top, so the steam can vent. Use a pastry brush to brush the beaten egg all over the top crust. Bake for 35-45 minutes, or until the crust is golden brown. Use a pie shield if you'd like to prevent the edges of the crust from browning too much too quickly. I put it on before putting the pie in the oven and take it off with about 10 or so minutes left of bake time.
Allow it to cool for 10 minutes before slicing and serving. Refrigerate any leftovers.
