Heat a large Dutch oven or pot over medium heat. Add the butter and heat until melted. Add the onion, carrots, and celery and cook, stirring frequently, for 6-7 minutes until the onions turn translucent.
Add the mushrooms, garlic, thyme, and rosemary, and cook for 5 minutes or until the mushroom juices have evaporated and the mushrooms turn golden.
Stir in the all-purpose flour and cook, stirring frequently, for 2 minutes until the flour smells nutty.
Stir in the chicken broth, salt, black pepper, and potatoes until well combined. Increase the heat to medium-high, bring the soup to a simmer, reduce to medium, and cook for 12-14 minutes or until the potatoes are fork-tender.
Stir in the chicken, peas, and corn. Bring back to a simmer and cook for 5 minutes.
Stir in the half-and-half and chopped parsley and cook for 1-2 minutes until heated through.
Season the soup to taste with salt and pepper and serve hot, garnished with chopped parsley and the cheesy puff pastry cutouts on top.