Heat a large, deep skillet over medium heat. Add the butter and melt.
Whisk in the flour and continue whisking over medium heat for 1 minute.
Slowly pour in the chicken broth and milk, whisking constantly. Continue cooking until mixture has thickened. Stir in salt and pepper.
Add the potatoes and carrots to a microwave safe bowl, cover with plastic wrap, and microwave for 3 minutes.
Add the chicken, potatoes, carrots, peas, and onion to a large bowl. Pour in the chicken broth mixture and stir to combine.
Arrange the bottom pie crust in the pie plate. Pour the filling over the crust. Add the top crust.
Crimp the edges and cut 4 small slits in the center of the top crust to vent.
Bake for 30-40 minutes or until the crust is golden brown.
Let sit for 5 minutes before slicing and serving.
After hearing from a couple readers that their carrots and potatoes weren't quite cooking through, I added an extra step of microwaving them before adding to the pot pie. You could also parboil them on the stove.