Heat the oven to 425°F and line an 18-by-13-inch rimmed sheet pan with foil. Toss chicken with 1 tablespoon neutral oil, salt and pepper and spread on one half of the pan; roast for 10 minutes.
Toss broccolini and red pepper with remaining neutral oil. Add them to the open side of the pan and turn the chicken pieces.
Roast for 10 minutes more. Stir honey, soy sauce, vinegar, garlic and sesame oil together in a clean bowl.
Spoon the glaze over chicken and vegetables and turn them to coat lightly. Roast for 5 to 7 minutes until chicken reaches 165°F, stems are tender and glaze is tacky rather than scorched.
Divide among 4 plates and spoon over the small amount of cooked pan sauce.