Preheat the oven to 375°F (190°C).
In a large oven-safe skillet or casserole dish, heat the olive oil and butter over medium heat until melted.
Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Add the chicken pieces to the skillet and season with salt, black pepper, paprika, and dried thyme. Cook for 5-6 minutes, stirring occasionally, until the chicken is lightly browned but not fully cooked.
Add the rinsed rice to the skillet and stir to combine with the chicken and onions.
Pour in the chicken broth, heavy cream, lemon juice, and lemon zest. Stir well to combine all ingredients.
Bring the mixture to a gentle simmer over medium heat.
Remove the skillet from heat and stir in the sour cream and shredded cheddar cheese until evenly mixed.
Sprinkle the grated Parmesan cheese evenly over the top of the casserole.
Cover the skillet or casserole dish tightly with aluminum foil.
Bake in the preheated oven for 35 minutes.
After 35 minutes, remove the foil and bake uncovered for an additional 10 minutes, or until the rice is tender and the top is golden and bubbly.
Remove from the oven and let the casserole rest for 5 minutes.
Sprinkle chopped fresh parsley over the top before serving.