In a small saucepan, combine the balsamic vinegar, honey, Dijon mustard, garlic, oregano, salt, and black pepper. Bring the mixture to a simmer over medium heat, stirring occasionally, until it has reduced by half and thickened to a glaze-like consistency, about 10-12 minutes. Remove the glaze from the heat and set it aside.
Preheat your oven to 400°F (200°C). In a large mixing bowl, toss the broccoli florets, baby carrots, sliced bell pepper, and onion with the olive oil. Season the vegetables with a pinch of salt and pepper, and arrange them in a single layer on a large baking sheet.
Place the chicken breasts on the baking sheet with the vegetables, making sure they are spaced out evenly. Brush the chicken generously with the prepared balsamic glaze, ensuring that all sides are coated.
Bake the chicken and vegetables in the preheated oven for 25-30 minutes, or until the chicken is cooked through and the vegetables are tender. Halfway through the baking time, baste the chicken with the remaining glaze from the baking sheet to keep it moist and flavorful.
