Remove stems from kale; wash leaves thoroughly. Bring 1 cup water to a boil in a large Dutch oven; add kale. Cook 15 minutes or until tender. Drain kale; cool slightly. Chop kale, and sprinkle with vinegar.
Heat oil in pan over medium heat. Add onion; sauté 3 minutes. Add garlic and mushrooms; sauté 2 minutes. Add rice; cook 1 minute, stirring frequently. Add basil, broth, and bay leaf. Bring to a boil; cover, reduce heat, and simmer 12 minutes. Spread kale over rice. Cover and cook 5 minutes or until liquid is absorbed. Discard bay leaf. Sprinkle with cheese and pepper; toss well.