Traditionally, a dish like this one might use preserved lemons. Since those are next to impossible to find (and pretty pricey!) I made my own "quick preserved lemons." They don't have quite the same flavor as the real deal, but they take less than a day and definitely serve their purpose in this recipe. Since the ingredients here are so simple, splurge on the best ones you can. I used a whole organic chicken that I broke down myself (it was under $8) and gorgeous Sicilian olives from the olive bar near my supermarket's deli counter.
Cut the lemons into a 1/4-inch dice (keep the peel on). In a small bowl, combine the lemons, 1 tablespoon salt, and sugar; mix well. Let sit at least 3 hours or cover and refrigerate up to 3 days.
Heat oven to 350ºF. Combine the ginger, turmeric, paprika, salt, and coriander. Rub onto the chicken skin. Heat the olive oil in a Dutch oven (or an ovenproof deep-sided skillet) set over medium heat. Add the chicken, skin-side down, and cook 8 minutes or until skin is golden and crisp. Flip the chicken over.
Add the garlic, potatoes, olives, carrot, and chicken stock. Stir into the salted lemons and any juice that accumulated in the bowl. Cover and transfer to the oven. Roast for 1 hour.