Prepare the filling: In a large saucepan, melt the butter over medium heat. Add the chopped onion and garlic, cooking until softened, about 3-4 minutes.
Make the sauce: Stir in the flour to form a roux, cooking for about 1 minute. Gradually whisk in the chicken broth and milk, stirring constantly until the mixture thickens.
Add the chicken and vegetables: Stir in the shredded chicken, mixed vegetables, thyme, rosemary, salt, and pepper. Cook for 5 minutes, allowing the flavors to meld together. Remove from heat.
Assemble the pie: Place one pie crust in a 9-inch pie dish. Pour the chicken mixture into the crust, spreading it evenly. Place the second pie crust over the top and seal the edges by pinching them together. Cut a few slits on top for ventilation.
Bake: Place the pie in the preheated oven and bake for 30-35 minutes, or until the crust is golden brown.
Cool and serve: Let the pie cool for about 10 minutes before slicing. Enjoy your homemade chicken pie!