Preheat your oven to 375°F (190°C). Grease a 9x13-inch baking dish or casserole dish.
In a large skillet, melt the butter over medium heat. Add the chopped onion and minced garlic, and sauté until softened, about 2-3 minutes.
Stir in the flour and cook for another 1-2 minutes to make a roux, stirring constantly.
Gradually whisk in the chicken broth and milk, stirring until the mixture thickens and is smooth.
Add the shredded or diced cooked chicken, mixed vegetables, dried thyme, salt, and pepper to the skillet. Stir to combine and cook for an additional 5 minutes, allowing the flavors to meld together.
Transfer the chicken and vegetable mixture to the prepared baking dish, spreading it out evenly.
Roll out the puff pastry sheet on a lightly floured surface to fit the size of your baking dish. Place the puff pastry over the filling, pressing the edges against the sides of the dish to seal.
Use a sharp knife to make a few slits in the puff pastry to allow steam to escape during baking. Brush the top of the pastry with beaten egg for a golden finish.
Bake in the preheated oven for 25-30 minutes, or until the puff pastry is golden brown and the filling is bubbling.
Remove from the oven and let the casserole cool for a few minutes before serving.