Preheat the Oven
Preheat your oven to 400°F (200°C).
Prepare the Filling
In a large skillet or Dutch oven, melt the butter over medium heat. Add the diced onion, garlic, carrots, and celery. Sauté until softened, about 5-7 minutes. Add the frozen peas and cooked chicken, stirring to combine.
Add the Sauce
Pour in the cream of chicken soup and chicken broth. Stir in the thyme, rosemary, salt, and pepper. Allow the mixture to simmer for 5 minutes until it thickens slightly. Remove from heat and pour the mixture into a 9x13-inch baking dish.
Make the Biscuit Topping
In a mixing bowl, whisk together the flour, baking powder, baking soda, salt, and fresh thyme leaves (if using). Using a pastry cutter or fork, cut in the cold butter until the mixture resembles coarse crumbs. Add the buttermilk and stir until a dough forms.
Top the Filling with Biscuits
Drop spoonfuls of the biscuit dough over the chicken filling, covering as evenly as possible. Leave a little space between the dough to allow it to spread as it bakes.
Bake
Place the baking dish in the preheated oven and bake for 20-25 minutes, or until the biscuit topping is golden brown and cooked through.
Serve
Let the cobbler cool for a few minutes before serving. Scoop generous portions onto plates and enjoy!