Preheat your oven to 425 degrees Fahrenheit.
In a large saucepan, melt the butter over medium heat. Add the onion and cook until soft and translucent, 3 to 5 minutes. Add the minced garlic and cook for another 30 seconds.
Add the flour, salt, pepper, thyme, and sage and whisk to combine. Cook for 1-2 minutes, until the mixture, is bubbly and golden.
Slowly pour in the chicken broth and milk, whisking constantly to prevent lumps from forming. Cook for 5-7 minutes, stirring frequently, until the mixture has thickened.
Add the cooked chicken, sliced mushrooms, frozen peas, carrots, and celery to the saucepan and stir to combine.
Roll out one of the pie crusts and place it in the bottom of a 9-inch pie dish. Pour the mixture into the prepared pie crust.
Roll out the second pie crust and place it on top of the filling. Crimp the edges to seal the pie.
Brush with the egg and cut a few slits in the top of the pie crust to allow steam to escape.
Bake the pie for 30-35 minutes, until the crust is golden brown and the filling is hot and bubbly.
Let the pie cool for a few minutes before slicing and serving. Enjoy!