In a large skillet over medium heat, melt the butter. Add the diced onion, celery, and minced garlic. Sauté for about 5 minutes until the vegetables are softened.
Sprinkle the flour over the sautéed vegetables and stir well to combine. Cook for another minute to eliminate the raw flour taste.
Gradually pour in the chicken broth and milk, stirring continuously to avoid lumps. Bring to a gentle simmer, allowing the mixture to thicken, about 3-5 minutes.
Stir in the shredded chicken and mixed vegetables. Add thyme, rosemary, salt, and pepper. Mix well and remove from heat.
Pour the chicken mixture into a large baking dish. Open the biscuit dough package and separate the biscuits. Layer the biscuits on top of the chicken mixture, making sure to slightly overlap them.
Place the baking dish in the oven and bake for 25-30 minutes, or until the biscuits are golden brown and cooked through.
Once done, remove from the oven and let it cool for a few minutes. Garnish with fresh parsley if desired. Serve hot and enjoy!