Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium-high heat.
Add the chicken pieces and cook until lightly browned and cooked through, about 5-7 minutes. Remove chicken from skillet and set aside.
In the same skillet, add diced onion and cook until translucent, about 3 minutes.
Add minced garlic, sliced carrots, broccoli florets, and mushrooms. Cook for 5 minutes until vegetables are slightly tender.
Sprinkle the flour over the vegetables and stir well to coat. Cook for 1-2 minutes to remove the raw flour taste.
Gradually pour in the chicken broth while stirring constantly to avoid lumps.
Add the milk, salt, black pepper, dried thyme, and dried parsley. Stir and cook until the sauce thickens, about 5 minutes.
Return the cooked chicken to the skillet and add frozen peas. Stir to combine.
Remove the skillet from heat and stir in shredded cheddar cheese until melted.
In a large mixing bowl, combine the chicken and vegetable mixture with the cooked rice. Mix well.
Transfer the mixture to a greased 9×13-inch baking dish.
Sprinkle grated Parmesan cheese evenly over the top.
Cover the dish tightly with aluminum foil and refrigerate if making ahead. When ready to bake, preheat the oven to 375°F (190°C).
Bake covered for 25 minutes, then remove the foil and bake for an additional 15-20 minutes until the top is golden and bubbly.
Remove from oven and let stand for 5 minutes before serving.