Preheat oven to 400°F (200°C). Line a large baking sheet with parchment paper or foil for easy cleanup.
Pat chicken breasts dry with paper towels. In a small bowl, mix 1 tsp salt, 1/2 tsp pepper, 1 tsp oregano, and 1/2 tsp garlic powder. Rub the seasoning all over the chicken. Drizzle with 1 tbsp olive oil.
In a large bowl, combine cherry tomatoes, bell peppers, red onion, and zucchini. Drizzle with remaining 2 tbsp olive oil, a pinch of salt and pepper, and toss to coat.
In a small bowl, whisk together balsamic vinegar, honey, minced garlic, Dijon mustard, and 1 tbsp olive oil until smooth.
Arrange the seasoned chicken breasts in the center of the prepared baking sheet. Scatter the vegetables around the chicken. Pour the balsamic glaze evenly over the chicken and vegetables, using a brush to coat the chicken well.
Bake for 25-30 minutes, or until chicken reaches an internal temperature of 165°F (74°C) and vegetables are tender and lightly caramelized. Halfway through, stir the vegetables and spoon pan juices over the chicken.
Remove from oven and let rest for 5 minutes. Serve directly on the baking tray or transfer to a platter. Spoon any remaining glaze from the pan over the chicken and veggies. Garnish with fresh basil or parsley if desired.