Remove all the skin and trim the fat from the chicken. Cut the chicken into 8 small pieces, using both the breast and the thighs, season with salt and pepper.
In a Dutch oven, heat 1 tablespoon of the oil over medium-high heat. When the oil starts to shimmer, add the chicken pieces and cook until golden brown, about 10 minutes on each side. Using a slotted spoon, remove the chicken pieces from the pan and set them aside. Add the onions and celery and let cook, stirring constantly, until soft and slightly golden, about 5 minutes. Return the chicken pieces to the pot. Add the currants, sugar, vinegar, olives, capers and half the olives. Season, to taste, with salt and pepper. Turn the heat to high and cook, turning the chicken constantly, until the vinegar evaporates, about 3 to 5 minutes. Turn the heat to medium-low, cover the saucepan and allow the chicken to cook through, about 20 to 25 minutes.
Serve family style, garnished with the remaining almonds, fennel fronds, juice from orange wedges and pepperoncini.