Heat olive oil in a large deep skillet or sauté pan over medium heat.
Add the chicken pieces and cook for 5-7 minutes until browned and cooked through. Remove chicken from the pan and set aside.
In the same pan, add diced onion and cook for 3 minutes until softened.
Add minced garlic, sliced mushrooms, and diced carrot. Cook for 5 minutes, stirring occasionally, until vegetables are 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 and heavy cream, stirring continuously until the sauce thickens, about 3-5 minutes.
Stir in dried thyme, salt, and black pepper.
Return the cooked chicken to the pan and add the frozen peas. Stir to combine and cook for an additional 5 minutes until heated through.
Add the cooked white rice and mix well to evenly distribute the sauce and ingredients.
Sprinkle shredded cheddar cheese evenly over the top, cover the pan with a lid, and cook for 2-3 minutes until the cheese melts.
Remove from heat and let the casserole rest for 2 minutes before serving.