Heat the oven to 375 degrees F. Massage salt and pepper all around the cavity of the chicken and on the skin of the bird. Drizzle with 2 tablespoons of the olive oil and rub that in as well. Set the chicken on a rack in a shallow roasting pan and roast until an instant-read thermometer inserted in the thickest part of the thigh reads 170 degrees to 175 degrees F, or the juices run clear, about 75 minutes. Remove from the oven and let cool to room temperature. Pick the meat off the chicken. By hand, shred the meat into bite-size pieces along the grain of the meat. Discard any skin, bones, or cartilage (or save for stock). If working ahead, wrap the meat tightly and refrigerate.

In a small saucepan, combine the remaining 3 tablespoons oil with the chopped garlic. Heat slowly until the garlic begins to brown, 4 to 5 minutes. Pour into a small bowl to cool. The garlic should be lightly toasted.

In a large mixing bowl, combine the shredded chicken with the mayonnaise and toss gently. Add the fennel, celery, olives, parsley, lemon zest, and toasted garlic with its oil. Drain the onions, dry them on a towel, and add them to the salad. If the salad seems dry, add more mayonnaise. Taste and adjust the seasonings. Put the salad on a platter or in a serving bowl. Cover with plastic wrap and refrigerate. Serve cold.