Combine water, brown rice, soy sauce, and salt in a rice cooker. Cover and cook according to manufacturer's directions, about 35 minutes.
Heat vegetable oil in a wide skillet over medium heat. Add curry paste; cook and stir until fragrant, 1 to 2 minutes. Add coconut milk, fish sauce, white sugar, and lime leaves. Reduce heat to medium-low and simmer, covered, until flavors combine, about 5 minutes. Stir in tofu, broccoli, red bell pepper, and carrots. Cook and stir until vegetables are tender but not soft, 1 to 2 minutes.