To Make Spice Mix: Combine first five ingredients in a small bowl. In a separate small bowl, cover olives with cold water to cover; set aside to soak.
In a large, deep-sided skillet or shallow braising pan, heat oil and butter over medium-high heat. Season chicken pieces lightly with salt. When oil and butter are hot, add half of the chicken (skin side down) and sear (without disturbing) until the skin is crisp and golden brown (about 5 minutes). Turn pieces over and repeat on this side. Remove pieces to a platter. Repeat the process with the other half of the chicken pieces. Remove pieces.
Pour off most of the liquid fats except for about 1 tablespoon. Return pan to medium heat and add onions and garlic. Stir with a wooden spoon and saute until you smell their delicious fragrances and they begin to soften (about 3 minutes). Add spice mix, stir, and saute for one minute longer. Then pour in 3/4 cup of water to deglaze the pan. Stir and scrape the bottom with the wooden spoon.
When water boils, return chicken pieces. Cover pan and reduce heat to low; braise the chicken for 10 minutes like this. Uncover pan, make sure liquid is at a gentle simmer, squeeze one-half of lemon over chicken, sprinkle half the herbs on top, return lid to cover, and braise gently for 10 minutes more.
While chicken is braising, prepare olives by draining them, then smashing them with the side of a chef's knife to remove the pits. Rinse preserved lemons, remove the pulp, and chop peel into small pieces.
After chicken has braised for a total of 20 minutes, lift lid, add olives and preserved lemons, and turn chicken pieces again. Replace lid and continue to braise for another 10 to 15 minutes.
Transfer chicken pieces to a serving dish or platter. Increase braising liquid to medium-high and bring to a boil. Squeeze other half of lemon over liquid. Reduce heat to a brisk simmer for about 5 minutes. Add remaining herbs. Season with salt and pepper. Spoon sauce over chicken to serve.
Note: Preserved lemons can be purchased at specialty stores but they are also very easy to make -you will just need to plan one day ahead.
Preserved Lemon Recipe: You will need organic lemons, coarse sea salt, paprika and extra-virgin olive oil. Wash and slice the lemons. Put the slices in a colander set over a bowl and sprinkle generously with salt. Leave overnight to drain off the moisture. Layer the lemon slices in a Mason jar and sprinkle each layer with paprika. Fill the jar to almost full, cover with extra-virgin olive oil and place in the refrigerator for about three weeks. They are worth the wait.