Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the chopped onion and cook until softened, about 3 minutes.
Add the minced garlic and sliced mushrooms to the skillet. Cook for 5 minutes until mushrooms release their moisture and soften.
Add the bite-sized chicken pieces to the skillet. Cook until the chicken is no longer pink on the outside, about 5 minutes.
Sprinkle the flour over the chicken and vegetables. Stir well and cook for 1 minute 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 until the sauce thickens, about 5 minutes.
Remove the skillet from heat and stir in the frozen peas, cooked white rice, and half of the shredded cheddar cheese.
Transfer the mixture to a greased 9×13-inch casserole dish. Sprinkle the remaining cheddar cheese, grated Parmesan cheese, and breadcrumbs evenly on top.
Bake in the preheated oven for 20-25 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.