Preheat oven to 350 degrees. Spray a 2 qt baking dish with nonstick cooking spray. Set aside.
In a large nonstick skillet, heat olive oil over medium heat. Add onion, garlic, mushrooms, salt and red pepper. Cook, stirring occasionally until onion softens, approximately 10 minutes. Add kale in batches, stirring until wilted after each addition. Stir in flour; cook 1 minute. Gradually add broth and half and half, stirring constantly until thickened, approximately 5 minutes. Add 1/2 c cheese and rice, stirring to combine. Spoon mixture into prepared dish. Sprinkle with remaining cheese.
Bake until bubbly and cheese melts, approximately 25 minutes. Serve immediately.