Step 1: Prepare the Filling

1. Heat olive oil or butter in a large skillet or oven-safe pan over medium heat.
2. Add the diced onion, garlic, and carrots. Sauté for 5-7 minutes until softened.
3. Stir in the shredded chicken, frozen peas, and frozen corn (if using). Cook for 2-3 minutes to combine.
4. Sprinkle the flour over the mixture and stir until evenly coated. Cook for 1-2 minutes to eliminate the raw flour taste.
5. Gradually pour in the chicken broth and milk, stirring constantly to create a smooth sauce.
6. Add thyme, smoked paprika, salt, and pepper. Bring to a simmer and cook for 5 minutes until thickened. Remove from heat and set aside.

Step 2: Make the Biscuit Topping

1. Preheat your oven to 400°F (200°C).
2. In a mixing bowl, whisk together the flour, baking powder, baking soda, and salt.
3. Add the cold butter cubes and use a pastry cutter or your fingers to work the butter into the flour mixture until it resembles coarse crumbs.
4. Pour in the buttermilk and mix gently until the dough comes together. Do not overmix.

Step 3: Assemble the Cobbler

1. Drop spoonfuls of the biscuit dough over the chicken filling, leaving small gaps for steam to escape.
2. Place the skillet or pan in the preheated oven and bake for 25-30 minutes, or until the biscuits are golden brown and cooked through.

Step 4: Serve and Garnish

1. Remove the Chicken Cobbler from the oven and let it cool for 5 minutes.
2. Garnish with fresh parsley and a sprinkle of Parmesan cheese, if desired.
3. Serve warm and enjoy!