In a large pot of boiling, salted water, cook the rice until just done, 10 to 15 minutes. Drain. Rinse with cold water. Drain thoroughly.
Coat the chicken with the 1 tablespoon oil. Season with 1/4 teaspoon each of the salt and pepper. Heat a grill pan over moderate heat. Cook the breasts until just done, 4 to 5 minutes per side. Alternatively, heat the tablespoon of oil in a large frying pan and season and cook the chicken as directed above. When the chicken is cool enough to handle, cut it into 1/2-inch dice.
Toss the rice with the chicken, onion, mango, avocado, the 1/3 cup oil, the remaining 1 teaspoon salt and 1/2 teaspoon pepper, the lime juice, and cilantro.
Wine Recommendation: With its fruit, richness, and flavor, Australian sémillon will match this dish blow for blow.