Heat olive oil in a large deep skillet or Dutch oven over medium heat.
Add the diced chicken pieces and cook for 5-6 minutes, stirring occasionally, until the chicken is lightly browned but not fully cooked through.
Add the diced onion and minced garlic to the skillet and cook for 3-4 minutes until the onion is softened and fragrant.
Pour in the chicken broth and add the uncooked rice, salt, black pepper, and dried thyme. Stir to combine.
Bring the mixture to a boil, then reduce heat to low and cover the skillet with a tight-fitting lid.
Simmer for 15 minutes without lifting the lid.
After 15 minutes, add the fresh broccoli florets on top of the rice and chicken mixture. Cover again and cook for an additional 5 minutes.
Remove the lid and stir the broccoli into the rice and chicken mixture.
Pour in the whole milk and stir to combine.
Add the shredded cheddar cheese, mozzarella cheese, and grated Parmesan cheese. Stir gently until all the cheese is melted and the casserole is creamy.
Cook uncovered for 2-3 more minutes to thicken the sauce slightly.
Remove from heat and let the casserole rest for 5 minutes before serving.