1. For the chicken: Mix all dry ingredients in a large bowl. Rinse the chicken well and pat dry.
2. Generously coat the chicken with the spice mixture, trying to get as much as possible directly on the meat, and then replacing the skin over-top.
3. To cook on the stovetop/oven, skip to step 6. To cook on the grill, preheat grill to high heat for about 15 minutes.
4. Leave outer burners on, and turn off the burners in the middle.
5. Put chicken on center part of grill grate, skin side down, and cook with the lid closed. Turn skin side up after 20 minutes, or when the skin is browned. Grill times may vary.
6. The chicken is fully cooked when the chicken breasts reach 165F at their thickest part, and the leg quarters reach 170F. This will probably take 40 minutes for the breasts and 50 minutes for the leg quarters.
7. To cook on the stovetop/oven, preheat oven to 375F. Heat oil in a large oven safe pan over medium-high heat.
8. Place the spice-rubbed chicken in the pan and cook, skin side down, until browned and crisp, about 20-30 minutes.
9. Turn over, transfer pan to hot oven, and roast until cooked through, 40-60 minutes.
10. Check the internal temperature of the chicken with a meat thermometer, looking for a minimum reading of 165F. Test the poultry at the thickest part, making sure not to touch the thermometer to a bone. Continue cooking the chicken until it reaches the minimum temperature before removing from the pan.
11. Transfer chicken to a large plate for serving.
12. For the zucchini: Increase grill heat to high or increase oven temperature to 450F. Slice zucchini lengthwise into half inch strips. Cut lemon into wedges. Roughly chop mint.
13. Drizzle zucchini with half of the oil, and season with salt and pepper.
14. To grill, place zucchini on the grill and cook for 4-6 minutes, turning often, until cooked through and tender. To roast, place zucchini on a lipped baking sheet for 15 minutes, or until zucchini is tender.
15. Transfer to a platter. Drizzle with remaining oil, squeeze lemon juice on top, and sprinkle with mint leaves
16. Serve zucchini alongside grilled chicken.