Heat a large skillet over medium-high heat; add oil and swirl to coat the skillet. Cook and stir chicken until browned, about 4 minutes. Add onion; cook and stir until softened, about 4 minutes. Add ginger and garlic and stir until fragrant, about 1 minute.
Pour chicken broth and rice wine into the skillet; scrape any browned bits off the bottom with a wooden spoon. Stir in honey, soy sauce, and Worcestershire sauce. Simmer sauce until thickened, about 5 minutes.
Stir 1/3 of the mustard greens into the skillet; cook and stir until wilted, 3 to 5 minutes. Repeat twice more with remaining mustard greens.