In a large non-stick skillet, melt butter over medium-high heat. Add potatoes, onions, celery, and carrots in the skillet. Add a pinch of salt and pepper to taste. Sauté for about 10 minutes or until the vegetables are tender. Stir in thyme leaves while cooking.
Sprinkle flour into the pan. Cook for one more minute, stirring constantly. Gradually stir in broth and milk. Reduce heat to medium and cook until the mixture is thickened and bubbly, stirring constantly.
Add turkey and stir well.
Pour the mixture into a 9-inch pie dish and top with pie crust. Trim off the excess crust along side the edge of your pie dish. Cut slits in the middle to allow steam to escape.
Bake the pie for 40-50 minutes or until pastry is golden brown and the filling is bubbly and cooked through.