Preheat your oven to 400°F (200°C) and line a large baking sheet with parchment paper.
Slice the zucchini into half-moons and chop the red bell pepper into bite-sized pieces.
In a large mixing bowl, toss the broccoli florets, zucchini, and bell pepper with half of the olive oil, the balsamic vinegar, and half of the salt and pepper.
Spread the vegetables in a single layer on the prepared baking sheet and roast for 20-25 minutes until tender and slightly caramelized at the edges.
While the vegetables are roasting, whisk together the remaining olive oil, lemon juice, minced garlic, and dried oregano in a small bowl.
Brush the chicken breast thoroughly with the lemon-herb mixture and season with the remaining salt and pepper.
Heat a grill or grill pan over medium-high heat and cook the chicken for 6-7 minutes per side, or until the internal temperature reaches 165°F.
Remove the chicken from the heat and let it rest for 5 minutes to lock in the juices before slicing.
Plate the sliced chicken alongside the roasted balsamic vegetables and serve immediately.