In a large oven-safe skillet over medium-high heat, heat oil. Add onion and cook, stirring, until soft, 5 minutes. Add garlic and cook until fragrant, 1 minute more. Add chicken and season with salt and pepper. Cook, stirring occasionally, until golden, about 6 minutes more.
Stir in rice, heavy cream, and 1 cup of the broth. Bring to a simmer and cook until rice is tender, about 15 minutes. Add remaining 1 1/2 cups broth, broccoli, and cheddar cheese and cook until broccoli is tender and cheese is melty, about 10 minutes.
Heat broiler. Sprinkle chicken mixture with bread crumbs and season with salt and pepper. Broil until golden and crispy, about 2 minutes.