Combine 4 teaspoons of the garlic, the paprika, cumin, ginger, coriander, cinnamon, and cayenne together in a small bowl; set aside. Mince 1 strip of the lemon zest and mix with the remaining teaspoon of minced garlic and 1 tablespoon of the honey in a separate small bowl; set aside.
Pat the chicken dry with paper towels and season with salt and pepper. Heat the oil in a large Dutch oven over medium-high heat until just smoking. Brown half of the chicken on both sides, 5 to 8 minutes per side, reducing the heat if the pan begins to scorch. Transfer the chicken to a plate, leaving the fat in the pot. Return the pot to medium-high heat and repeat with the remaining chicken; transfer the chicken to the plate.
Pour off all but 1 tablespoon fat from the pot. (Add additional oil to equal 1 tablespoon, if needed.) Add the onion, the remaining 2 lemon zest strips, and 1/4 teaspoon salt to the pot and cook over medium heat, stirring occasionally, until softened, 5 to 7 minutes. Stir in the garlic-spice mixture and cook until fragrant, about 30 seconds. Stir in the carrots, broth, and the remaining 1 tablespoon honey, scraping up any browned bits.
Nestle the chicken, along with any accumulated juices, into the pot and bring to a simmer. Cover, turn the heat to medium-low, and simmer until the chicken is fully cooked and tender, about 20 minutes for the breasts (160 to 165 degrees on an instant-read thermometer), or 1 hour for the thighs and drumsticks (170 to 175 degrees on an instant-read thermometer). (If using both types of chicken, simmer the thighs and drumsticks for 40 minutes before adding the breasts.)
Transfer the chicken to a serving dish, tent loosely with foil, and let rest while finishing the sauce. Skim as much fat as possible off the surface of the sauce, add the figs, and return to a simmer until the sauce has thickened slightly and the carrots are tender, 4 to 6 minutes. Stir in the garlic/lemon-zest/honey mixture, cilantro, and lemon juice. Season with salt and pepper to taste. Pour the sauce over the chicken and serve.