1. Preheat the oven to 375°F.

2. Melt the butter in a large oven-safe skillet or deep pan over medium heat.

3. Add the onion, carrots, and celery, then cook until softened.

4. Stir in the garlic and cook for 30 seconds.

5. Sprinkle the flour over the vegetables and stir well to coat.

6. Slowly pour in the chicken broth while stirring to keep the mixture smooth.

7. Add the heavy cream, shredded chicken, potatoes, salt, black pepper, and dried thyme.

8. Simmer for 10 to 12 minutes, until the potatoes begin to soften and the filling thickens slightly.

9. Arrange the refrigerated biscuits on top of the hot filling.

10. Transfer the skillet or mixture in a baking dish to the oven.

11. Bake for 20 to 25 minutes, until the biscuits are golden brown and cooked through.

12. Let rest for 5 minutes, then sprinkle with chopped parsley and serve warm.