Heat oven to 425°F; foil-line 18-by-13-inch rimmed sheet. Toss chicken with 1 tablespoon neutral oil, salt, and pepper. Spread over half pan; roast 10 minutes.
Toss broccolini and pepper with remaining neutral oil. Add to open half; turn chicken.
Roast 10 minutes more. Mix honey, soy sauce, vinegar, garlic, and sesame oil in clean bowl.
Spoon glaze over chicken and vegetables; turn to lightly coat. Roast 5 to 7 minutes until chicken reaches 165°F, stems tender, and glaze tacky, not scorched.
Divide among 4 plates; spoon over small amount of cooked pan sauce.