Cut tangerines in half and squeeze juice by hand (to preserve the shape of the peels); set peels aside. In a medium bowl, combine tangerine juice (about 1/3 cup), cornstarch, sugar, sherry, and soy sauce. Add beef and chiles and marinate 30 minutes.
Meanwhile, bring 2 in. water to a boil in a small saucepan. Scrape out pulp and white pith from tangerine peels with a spoon and discard. Add peels to pan and cook, covered, 2 minutes. Drain. Cut into long 1/2-in.-wide strips (discard stem ends) and set aside.
With a slotted spoon, lift beef and chiles from marinade to a plate (reserve marinade). Pat beef and chiles fairly dry with paper towels.
Bring 1/2 in. fresh water to a boil in a 12-in. frying pan (not nonstick). Add broccolini, cover, and cook, stirring occasionally, until tender-crisp, 3 to 4 minutes. Drain, rinse with cold water until cool, and set aside.
Heat pan over medium-high heat until hot. Add 2 tbsp. oil and swirl to coat, then add half the ginger and tangerine strips. Stir-fry until strips start to turn golden in spots, 30 to 60 seconds. Increase heat to high. Add half the beef and chiles. Stir-fry until beef browns, 1 1/2 to 2 minutes. With a slotted spoon, transfer meat and seasonings to a clean plate. Repeat cooking with another 2 tbsp. oil and remaining ginger, tangerine strips, beef, and chiles; reduce heat if pan starts to scorch.
Add remaining 1 tbsp. oil, the garlic, and green onions to pan. Stir once or twice, then add broccolini and stir-fry until hot, about 1 minute.
Stir vinegar into reserved marinade. Add marinade and beef to pan and cook, stirring, until sauce bubbles, about 1 minute. Serve immediately, with rice, to keep colors bright.