Preheat the oven to 425°F (220°C) and line a large sheet pan with foil or parchment for easy cleanup.
On a cutting board, chop the chicken into bite-size pieces, cut the broccoli into florets, and chop the bell pepper and zucchini.
In a large bowl, combine the chopped chicken, broccoli florets, chopped bell pepper, chopped zucchini, and 1/4 cup basil pesto; toss until everything is evenly coated.
Spread the mixture in a single layer on the prepared sheet pan so pieces roast rather than steam.
Bake for 15–20 minutes, until the chicken is cooked through (no pink inside) and the vegetables are tender-crisp.
If using mozzarella, sprinkle or drizzle the 1/2 cup over the pan during the last 5 minutes of baking so it melts, then remove from oven and let rest a couple minutes before serving.
Serve warm alongside pasta, rice, or toasted bread if desired.