Preheat oven to 400°F (200°C). Remove pie crusts from refrigerator and let sit at room temperature for 15 minutes. Lightly grease a 9-inch pie dish with butter or non-stick spray. Place a baking sheet on the lower oven rack to catch drips.
In a large bowl, whisk together cream of chicken soup, chicken broth, and milk until completely smooth. Add garlic powder, onion powder, thyme, rosemary, salt, and pepper, then whisk again.
Stir in the shredded chicken until well coated. Add frozen mixed vegetables and stir gently to combine. Taste and adjust seasoning if needed.
Unroll one pie crust and press it into the bottom and sides of the prepared pie dish. Trim excess, leaving a 1/2-inch overhang. Pour the filling into the crust-lined dish and spread evenly.
Unroll the second pie crust and place it over the filling. Press edges together to seal, trim excess, fold overhang under, and crimp decoratively. Cut 4-5 small slits in the center for steam to escape. Brush with egg wash if desired.
Bake on the middle rack for 35-40 minutes until crust is deep golden brown and filling is bubbling through the vents. If edges brown too quickly, cover loosely with foil. Remove from oven and let rest for 15-20 minutes before serving.