Preheat oven to 400°. In a small bowl, whisk together 1/4 cup water, the vinegar, and cornstarch. Whisk in honey and set aside. Sprinkle chicken on both sides with 3/4 tsp. salt and 1/2 tsp. pepper.
Heat a heavy 12- to 14-in. ovenproof frying pan over medium-high heat until hot. Add oil and swirl to coat, then add chicken skin side down and brown until skin is deep golden and crisp, 8 to 10 minutes. Turn over and brown other side, 2 to 3 minutes more. Transfer chicken to a plate and set aside.
Discard all but 2 tbsp. fat from frying pan, reduce heat to medium, and cook shallots until translucent, 3 to 5 minutes. Add garlic and cook 1 minute more. Add cornstarch mixture, chopped oregano, and remaining 1/4 tsp. each salt and pepper; cook until mixture bubbles and thickens, 1 to 2 minutes.
Set chicken skin side up in frying pan and add figs. Roast in oven until figs soften and caramelize slightly and meat is no longer pink at bone, about 15 minutes. Divide chicken, figs, and sauce among plates and top with whole oregano leaves.