In a 2- to 3-quart pan, combine rice and 2 3/4 cups water. Bring to a boil over high heat and cook until most of the water is absorbed, 7 to 10 minutes. Reduce heat to low, cover, and simmer until rice is tender to bite, 10 to 15 minutes longer.
Cut the lamb into strips about 1/4 inch thick.
In a 5- to 6-quart pan over medium heat, combine oil, onion, and garlic; stir often until onion begins to brown lightly, 6 to 7 minutes.
Add curry powder and cumin and stir until spices are fragrant, about 30 seconds.
Stir in meat. Add 1 1/2 cups broth and bring to a boil over high heat. Mix cornstarch and 1/4 cup broth. Stir into pan and stir until boiling resumes. Add salt and cayenne to taste. Spoon curry into a bowl.
Place condiments - banana, yogurt, chutney, cucumber, and currants - in small bowls. Serve curry with rice and condiments to add to taste.
Note: Cooked chicken can be substituted for the lamb; increase total broth to 2 1/2 cups.