Preheat the oven to 400°F. Melt the butter in a large skillet over medium heat.
Add the diced onion, carrots, and celery. Cook for 6 to 8 minutes, stirring occasionally, until the vegetables have softened.
Sprinkle the flour over the vegetables and stir constantly for 1 minute to cook out the raw flour taste.
Gradually whisk in the chicken broth, followed by the milk. Continue stirring until the sauce thickens and becomes smooth.
Stir in the cooked chicken, salt, black pepper, and dried thyme. Simmer for 2 to 3 minutes until everything is heated through.
Fit one refrigerated pie crust into a 9-inch pie dish. Spoon the chicken filling evenly into the crust.
Place the second pie crust over the filling. Trim and crimp the edges to seal, then cut several small slits in the top crust to allow steam to escape.
Bake for 35 to 40 minutes, or until the crust is golden brown and the filling is bubbling around the edges.
Let the pie cool for 15 minutes before slicing. Garnish with fresh thyme if desired and serve warm.