Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the chicken pieces and cook until lightly browned but not fully cooked, about 4-5 minutes. Remove chicken and set aside.
In the same skillet, add diced onion and minced garlic. Sauté for 3 minutes until softened.
Add diced carrots and sliced mushrooms to the skillet and cook for another 5 minutes until vegetables begin to soften.
Sprinkle the flour over the vegetables and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the chicken broth and milk, stirring constantly to avoid lumps. Bring the mixture to a simmer and cook until thickened, about 5 minutes.
Stir in dried thyme, dried parsley, salt, and black pepper.
Return the browned chicken to the skillet. Add frozen peas and corn. Stir to combine and cook for 2 minutes.
Remove the skillet from heat and fold in the cooked white rice.
Transfer the mixture to a greased 9×13 inch casserole dish. Sprinkle shredded cheddar cheese evenly over the top.
Cover the casserole with aluminum foil and bake in the preheated oven for 25 minutes.
Remove the foil and bake for an additional 10-15 minutes until the cheese is melted and bubbly.
Allow the casserole to cool slightly before serving. Store leftovers in an airtight container in the refrigerator.