Arrange a rack in center of oven; preheat to 375°. Pat chicken dry with paper towels; season with salt and pepper. In a large skillet over medium heat, heat oil until shimmering. Cook chicken, turning occasionally, until deeply golden brown, about 5 minutes per side. Transfer to a plate. Let cool slightly, then shred with 2 forks.
In same skillet over medium-low heat, add onion; season with salt. Cook, stirring occasionally, until softened and just turning brown, about 10 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Increase heat to medium and add rice. Cook, stirring occasionally, until rice is translucent, about 1 minute.
Transfer rice mixture to a 13” x 9” baking dish. Add soup and broth; season with salt and pepper, if needed, and stir well to combine.
Cover dish with foil and bake 1 hour. Uncover, fold in shredded chicken, then cover again with foil. Continue to bake until rice is cooked through, about 10 minutes more.
Turn on broiler. Top casserole with cheese. Broil, watching closely, until cheese is crispy and golden, 1 to 2 minutes. Top with parsley.
