Cut tangerines in half and squeeze juice by hand (to preserve the shape of the peels) to make about 2/3 cup; discard seeds. Set peels aside. In a medium bowl, combine tangerine juice, vinegar, soy sauce, honey, cornstarch, 1/4 tsp. salt, and 3/4 tsp. pepper.
With tip of a spoon, scrape out pulp and white pith from inside of tangerine peels and discard. Bring 2 in. water to a boil in a small saucepan. Add peels to pan, reduce heat, and simmer, covered, until just tender, 5 minutes. Drain. Quarter peels and set aside.
Meanwhile, preheat oven to 400°. Sprinkle chicken on both sides with remaining 1 tsp. salt and 1/2 tsp. pepper.
Heat a heavy 12-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, 5 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 1 tbsp. fat from frying pan and re- duce heat to medium. Add garlic and ginger and cook, stirring, until softened, about 30 seconds. Add tangerine peels and sauce mixture; cook, stirring, until mixture bubbles and thickens, about 2 minutes.
Set chicken, skin side up, in frying pan (don’t spoon sauce on top, or you’ll lose the crisp skin). Roast in oven until meat is no longer pink at bone, 10 to 15 minutes. Scatter green onion and basil over chicken. Toss gently and serve immediately.
*Clementines may also be sold as Cuties, Sweeties, Sunkist Delites, and Halos.