Step 1: Prepare the Filling

1. Melt the butter in a large skillet or saucepan over medium heat. Add the onion and garlic, sautéing until softened and fragrant, about 3 minutes.
2. Stir in the carrots, cooking for another 5 minutes until slightly tender.
3. Sprinkle the flour over the vegetables, stirring constantly to coat them evenly. Cook for 1-2 minutes to remove the raw flour taste.
4. Gradually add the chicken broth, whisking to combine. Then, slowly stir in the milk or cream. Simmer for 5-7 minutes, or until the mixture thickens into a creamy sauce.
5. Add the shredded chicken, peas, corn, thyme, and rosemary. Season with salt and pepper to taste. Stir well and remove from heat.

Step 2: Assemble the Casserole

1. Preheat your oven to 400°F (200°C). Grease a 9×13-inch baking dish.
2. Pour the chicken and vegetable filling into the prepared baking dish, spreading it out evenly.

Step 3: Add the Crust

1. Roll out the puff pastry (or biscuit dough) to fit the size of your baking dish. Lay it over the filling, tucking in the edges slightly.
2. Brush the pastry with the beaten egg to create a golden, shiny finish. Use a sharp knife to cut a few small slits in the crust to allow steam to escape.

Step 4: Bake the Casserole

1. Place the casserole on the center rack of the preheated oven. Bake for 30-35 minutes, or until the crust is golden brown and flaky.
2. Let the casserole cool for 5-10 minutes before serving.

Step 5: Garnish and Serve

1. Sprinkle fresh parsley over the top for a burst of color and flavor. Serve hot with a side salad or crusty bread.