Place the chicken in a large pot with water to cover. Add the salt and bring to a boil, then reduce the heat to a low simmer. Cook until the chicken is cooked through and tender, about 30 minutes. Drain the chicken, cover and refrigerate until cool or up to 24 hours.
In a large bowl, stir together the mayonnaise, olive oil, orange juice and some salt and pepper. Dice the cooled chicken, then add it to the dressing along with the rice, grapes, pineapples and almonds, folding gently to coat the ingredients with the dressing. Cover the bowl and refrigerate for at least 2 hours or up to overnight to allow the flavors to develop.
Serve chilled in endive cups.
Adapted from "Georgia Cooking in an Oklahoma Kitchen" by Trisha Yearwood © Clarkson Potter 2008. Provided courtesy of Trisha Yearwood. All rights reserved.