1. Preheat oven to 375°.
2. Bring broth and water to a boil in a medium saucepan. Slowly stir in grits and 1/2 teaspoon pepper; cover, reduce heat, and simmer 20 minutes, stirring occasionally.
3. Remove grits from heat. Combine milk and eggs in a small bowl, stirring with a whisk; stir into grits with whisk. Stir in cheese.
4. Rub a 2-quart casserole with cut sides of halved garlic clove; pour grits into prepared dish. Bake at 375° for 30 minutes. Let stand 10 minutes before serving (mixture will be dense and puddinglike when set).
5. While grits bake, bring 2 quarts water to a boil in a large saucepan. Add kale; reduce heat, and simmer 5 minutes. Remove kale from pan; drain and rinse under cold water.
6. Add olive oil, onion, chopped garlic, and salt to pan; sauté 2 minutes or until onion is translucent. Return kale to pan, and sauté 10 minutes or until greens are tender. Stir in remaining 1/2 teaspoon pepper and balsamic vinegar. Serve greens over grits; sprinkle with goat cheese.