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Lightly season chicken all over with sea salt and pepper. In large Dutch oven or other heavy pot over medium-high, heat vegetable oil. Sear thighs 8–10 minutes, skin side down, until skin is browned and crisp. Transfer to plate, skin side up.
Add shallots and ginger to pot, stirring 3 minutes, until golden. Add wine, bring to simmer, and cook 2 minutes or until reduced to half. Add soy sauce, brown sugar, sesame oil, and 1/2 cup broth and bring to simmer, stirring to dissolve sugar. Return chicken to pot, placing skin side up, and simmer 25–30 minutes until chicken is cooked through. Transfer chicken to a plate.
Add carrots, mustard greens and remaining broth to pot. Make sure carrots are mostly submerged. Partially cover, and simmer 10–12 minutes, until carrots are barely fork-tender and greens are wilted. Uncover, increase heat to medium, and continue to cook 5-10 minutes until liquid is reduced.
Remove pot from heat and drizzle vinegar over vegetables. Taste sauce and adjust seasoning, if needed. Put chicken back in pot and turn to coat in sauce. Serve with rice, if desired.