In a large oven-safe skillet heat oil over medium-high heat. Add onion and cook until soft, about 5 minutes. Add garlic and cook 1 minute more. Add chicken and season with salt and pepper. Cook, stirring occasionally, until chicken is golden on the outside, about 6 minutes.
Stir in rice, heavy cream, and chicken broth. Bring to a simmer over medium low heat, cover and cook until rice is tender (about 30 minutes), stirring occasionally.
When rice is tender, add broccoli, and 1½ cups of the cheddar cheese. Mix until combined and continue simmering until broccoli is tender and cheese is melted, about 5 minutes. Season with more salt and pepper if needed and add more chicken broth if needed. (You want this mixture to be moist and more on the “wet” side.)
In a small bowl, mix together the bread crumbs, parmesan cheese and melted butter, set aside. Sprinkle the remaining ½ cup grated cheese over chicken & broccoli followed by the bread crumb mixture. Broil until golden and crispy, about 2 minutes. Serve immediately.