Preheat the oven to 375°F (190°C).
In a large skillet, melt butter over medium heat.
Add chopped onions and garlic; sauté until translucent.
Stir in the flour and cook for another minute.
Gradually whisk in chicken broth and then milk.
Add thyme, salt, and pepper; bring to a simmer.
Stir in shredded chicken and mixed vegetables; cook until heated through.
Transfer the mixture into a baking dish and spread evenly.
Bake in the preheated oven for 25-30 minutes until bubbly.
Let it cool for a few minutes before serving.