In a large skillet, melt the butter over medium heat. Add the diced onion, celery, and minced garlic. Sauté for about 5 minutes until the vegetables are soft and fragrant.
Sprinkle the flour over the sautéed vegetables, stirring well. Cook for an additional minute to eliminate the raw flour taste.
Gradually pour in the chicken broth and milk while stirring continuously. Bring to a gentle simmer and allow to thicken for about 3-5 minutes.
Stir in the shredded chicken and mixed vegetables. Add thyme, rosemary, salt, and black pepper. Mix well and remove from heat.
Pour the chicken and vegetable mixture into a large baking dish. Open the biscuit dough package and layer the biscuits over the mixture, slightly overlapping them.
Bake in the preheated oven for 25-30 minutes until the biscuits are golden brown.
Let cool for a few minutes before serving. Garnish with freshly chopped parsley if desired.
Notes
Serve with a side salad or crusty bread to soak up the sauce.