Preheat the oven to 375°F (190°C). Lightly grease a 9×13 inch casserole dish.
In a large skillet, melt the butter over medium heat. Add the chopped onion and cook until softened and translucent, about 5 minutes.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Stir in the uncooked rice and cook for 2 minutes, stirring frequently to coat the rice with butter and toast slightly.
In a medium bowl, whisk together the chicken broth, whole milk, sour cream, lemon juice, lemon zest, dried thyme, salt, and black pepper until smooth.
Pour the liquid mixture into the skillet with the rice and onions. Stir to combine and bring to a gentle simmer.
Remove the skillet from heat and fold in the cooked chicken and half of the cheddar cheese.
Transfer the mixture into the prepared casserole dish and spread evenly.
Sprinkle the remaining cheddar cheese and the Parmesan cheese evenly over the top.
Cover the casserole dish tightly with aluminum foil and bake for 35 minutes.
Remove the foil and bake for an additional 10 minutes, or until the cheese is golden and bubbly and the rice is tender.
Remove from the oven and let the casserole rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.