Preheat oven to 425°F. Toss Brussels sprouts with 1 tbsp olive oil, salt, and pepper; spread on a baking sheet.
Season chicken with salt and pepper and drizzle with remaining olive oil. Place on baking sheet with sprouts.
Roast for 18–22 minutes until chicken reaches 165°F and sprouts are browned at edges.
Meanwhile, simmer balsamic, honey, mustard, garlic, and red pepper flakes in a small saucepan until reduced by half, about 6–8 minutes.
Stir in butter off heat for shine, then brush glaze over chicken and sprouts. Let rest 5 minutes before serving.