Heat the olive oil in a large skillet over medium heat. Pat dry the chicken pieces and season well with salt and pepper.
Place the chicken pieces skin-side down in the skillet and cook until the skin is golden brown, about ten minutes. Turn and brown the other side.
Remove chicken to a plate. Pour off all but 2 tablespoons of the fat in the pan.
Add the shallots, garlic and preserved lemon to the pan and sauté over medium heat until softened, about five minutes.
Add the cinnamon and allspice and sauté a few additional minutes until fragrant.
Add the chicken broth to the pan and, using a wooden spoon, scrape up any browned bits on the bottom of the skillet.
Return the chicken pieces to the pan, arranging them in a single layer. Add the figs and thyme and raise the heat to high.
Bring the liquid to a boil, then cover the skillet and turn heat down to low. Simmer until chicken is cooked through and registers 165 degrees on a meat thermometer, about 20 minutes.
While the chicken is cooking, toast the almonds in a dry skillet for a few minutes until golden and fragrant - watch carefully so that they don't burn.
To serve, arrange the chicken and figs on a platter. Spoon sauce over chicken and top with toasted almonds. Garnish with a few additional sprigs of thyme if desired.
Note: A staple of Middle Eastern cooking, preserved lemons add a unique pickled tanginess to North African-inspired dishes such as this one. You can find jars of preserved lemons in good grocery stores or Middle Eastern markets or make your own. If you are not able to source preserved lemons, you can substitute cooked fresh lemon. Combine one thinly sliced lemon with a half cup of fresh lemon juice and two teaspoons of kosher salt in a small saucepan. Bring to a boil, stirring to dissolve the salt, then cover and reduce heat to low. Simmer until lemon slices are tender, about ten minutes. Cool and chop.