Rice Salad

This is a recipe I got from an old neighbor who was from Puerto Rico.  Everyone always begged her for this recipe (including me!).  It sounds a little different, but is delicious! 

• 1 1/2 cups rice, preferably short grain
• 1 1/3 pounds boneless, skinless chicken breasts (about 4)
• 1/3 cup plus 1 tablespoon cooking oil
• 1 1/4 teaspoons salt
• 3/4 teaspoon fresh-ground black pepper
• 3/4 cup chopped red onion
• 1 banana, peeled and chopped
• 1 avocado, peeled and chopped*
• 3 1/2 tablespoons lime juice (from about 2 limes)
• 3/4 cup chopped cilantro

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. 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 pieces.
Toss the rice with the chicken, onion, banana, avocado, the 1/3 cup oil, the remaining 1 teaspoon salt and 1/2 teaspoon pepper, the lime juice, and cilantro.