Preheat oven to 400°F.
Gently wash produce under cool running water, pat dry, and prepare as directed.
Spray the rimmed baking sheet with nonstick cooking spray, or line it with parchment paper.
Toss the halved potatoes with ½ Tbsp olive oil, ¼ tsp salt, and ¼ tsp black pepper. Spread on the baking sheet, leaving about one third of the pan empty.
Season the chicken with ½ Tbsp olive oil, the minced garlic, and a dash each of salt and pepper. Add to the baking sheet.
Bake for 30 minutes.
Remove the pan from the oven and add the zucchini to the empty portion of the pan. Season the zucchini with the remaining salt and pepper.
Sprinkle the Parmesan cheese over the vegetables and chicken.
Bake for an additional 10–15 minutes, until the vegetables are tender and the chicken reaches 165°F when checked with a food thermometer.