In a small saucepan, cover the rice with the water and bring to a boil. Reduce the heat, cover and simmer until the rice is tender, 35 to 45 minutes. Remove from the heat and let stand for 10 minutes, then season with salt and fluff with a fork.
Meanwhile, in a large saucepan, combine the chicken, scallions, garlic, corn, cumin and 1/2 cup of the cilantro with the chicken broth. Season with salt and pepper and bring to a boil. Simmer the broth over moderately high heat until the chicken is cooked through, about 12 minutes. Transfer the chicken to a plate and let cool slightly. Pull the meat off the bones and shred.
Strain the broth and return it to the saucepan. Return the corn to the broth and discard the remaining solids. Bring the broth to a boil. Add the potatoes and simmer over moderately high heat until nearly tender, about 8 minutes. Add the asparagus and simmer until the potatoes and asparagus are tender, about 5 minutes longer. Return the shredded chicken to the pot and season the soup with salt and pepper.
Ladle the soup into bowls and garnish with the avocado, yogurt, capers, brown rice and remaining 2 tablespoons of cilantro.