Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and sliced mushrooms to the skillet. Cook for 5-6 minutes until mushrooms release their moisture and begin to brown.
Add the chicken pieces to the skillet and cook for 5-7 minutes until the chicken is no longer pink on the outside.
Stir in the diced carrots and frozen peas. Cook for 2 minutes to combine flavors.
Sprinkle the flour over the mixture and stir well to coat all ingredients. Cook for 1-2 minutes to remove the raw flour taste.
Gradually pour in the chicken broth while stirring continuously to avoid lumps.
Add the heavy cream, dried thyme, dried parsley, salt, and black pepper. Stir and cook for 3-4 minutes until the sauce thickens.
Remove the skillet from heat and stir in the shredded cheddar cheese until melted and combined.
Transfer the chicken and sauce mixture into a 9×13 inch casserole dish.
In a small bowl, combine the breadcrumbs, grated Parmesan cheese, and melted butter. Mix until the breadcrumbs are evenly coated.
Sprinkle the breadcrumb mixture evenly over the top of the casserole.
Bake in the preheated oven for 25-30 minutes or until the topping is golden brown and the casserole is bubbly.
Remove from the oven and let it rest for 5 minutes before serving.