Place potatoes in a large saucepan. Cover with cold, salted water, and bring to a boil. Reduce heat and simmer until very tender, about 20 minutes. Let cool.
Peel potatoes and pass through a food mill or ricer (or simply mash very finely) into a large bowl. Knead lightly with gloved hands (to avoid getting blue color on skin), slowly drizzling in olive oil, as needed, until potatoes reach a doughlike consistency. Add lime juice, and season to taste with salt. Refrigerate until cold and firm, about 2 hours.
Put chicken, onion, carrot and mint into a large saucepan, adding just enough water to cover; bring to a slow boil. Cook until chicken is fork-tender and can be pulled apart, about 20 minutes.
Toss or compost whole onion, carrot and mint. Transfer chicken to a medium bowl. Once cool enough to handle, shred chicken with fingers or a fork. Mix in yogurt, jalapenos, celery and red onion. Season to taste with salt and pepper. Refrigerate until cold, about 1 hour.
Oil a 2-quart casserole. Spoon half of the potato mixture into the casserole, followed by the chicken mixture, and then the remaining potato mixture. Refrigerate until firm, about 2 hours.
Put a dollop of avocado dressing on each plate. Slice and serve on dressing; garnish with avocados and olives.