Heat olive oil in a large deep skillet or Dutch oven over medium-high heat.
Add the chicken pieces and cook for 5-6 minutes, stirring occasionally, until lightly browned but not fully cooked through. Remove chicken and set aside.
In the same pot, add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 30 seconds until fragrant.
Stir in the rinsed rice and cook for 1 minute to lightly toast.
Pour in the chicken broth and milk, then add salt, black pepper, dried thyme, and smoked paprika. Stir to combine.
Return the chicken pieces to the pot and bring the mixture to a boil.
Reduce heat to low, cover the pot, and simmer for 15 minutes.
After 15 minutes, stir in the broccoli florets, cover again, and cook for an additional 8-10 minutes until the rice is tender and broccoli is bright green and slightly tender.
Remove the pot from heat and sprinkle shredded cheddar, mozzarella, and Parmesan cheeses evenly over the top.
Cover the pot and let it sit for 5 minutes to allow the cheese to melt.
Gently stir the casserole to combine the melted cheese evenly before serving.