Heat the oven to 400°F (200°C).
Melt butter in a saucepan over medium heat. Add onion and cook until softened, about 4 minutes. Stir in flour and cook for 1 minute.
Whisk in broth and milk gradually. Add thyme, salt, and pepper; simmer 3 to 5 minutes until thickened. Stir in chicken and frozen vegetables. Let the filling cool for 10 minutes.
Fit one pie crust into a 9-inch pie plate. Add the filling, cover with the second crust, seal and crimp the edge, then cut several vents in the top. Beat egg with water and brush over the crust.
Bake 35 to 40 minutes until the crust is deep golden and the center of the filling reaches 165°F (74°C). If the edges brown too quickly, cover them loosely with foil. Rest 15 minutes before serving.