To prepare the Moroccan Spice Mix for the Chicken: Combine the cinnamon stick, cloves, cayenne, cumin, fennel, coriander, and paprika in a dry skillet over low heat and toast for just a minute to release the fragrant oils; shake the pan so they don't scorch. In a spice mill or clean coffee grinder, grind the toasted spices together, with 1 1/2 teaspoons of kosher salt and the brown sugar. Preheat the oven to 400 degrees F.
Rinse the chicken with cool water, inside and out, then pat it dry with paper towels. Massage the chicken skin with the spice rub; make sure you don't miss a spot. Season the inside of the chicken generously with salt and pepper. Stuff the lemons halves, cilantro, and garlic in the cavity and place the chicken in a roasting pan fitted with a rack. Fold the wing tips under the bird and tie the legs together with kitchen string. Drizzle the oil all over the chicken. If you have time, let the chicken marinate for 30 minutes to really get the flavors down deep into the meat. Roast the chicken for 1 hour; pop an instant-read thermometer into the thickest part of the thigh; if it reads 160 degrees F, it's done. Allow the chicken to rest for 10 minutes so the juices can settle back into the meat. Remove and discard the skin from the chicken. Pull the chicken from the bone and shred the meat, with your fingers or 2 forks. Put the shredded chicken in a large bowl and squeeze the lemon halves that have cooked inside the bird over the meat to moisten.
To prepare the Apricot Couscous: Put the couscous in a medium bowl; pour the water over it, stir with a fork to combine. Cover and let sit for 10 to 15 minutes, then uncover and fluff with a fork. Put the apricots, almonds, green onions, mint, and cilantro on a cutting board and coarsely chop everything up; add this to the couscous. Add lemon juice, drizzle with olive oil, and season with salt and pepper. Toss gently to combine.
To prepare the Green Olive Sauce: In a small skillet, heat 2 tablespoons of olive oil. Saute the shallots and chile over low heat for 8 minutes until caramelized, then scrape into a food processor. Add the olives, parsley, vinegar, lemon juice, and 1/2 cup of oil; puree a good 3 minutes until totally smooth.
To put the whole thing together: put the shredded chicken and couscous on the lavosh bread, add a few spoonfuls of the olive sauce, and roll the sandwich up. Wrap in foil and take the sandwich to a picnic, if desired.