Preheat oven to 400°F (200°C).
In a large mixing bowl, whisk together pesto, olive oil, ranch seasoning, minced garlic, Dijon mustard, chili powder, salt, and pepper to make the pesto mix.
Line a 21×15-inch jelly roll pan with foil and lightly spray with cooking spray.
Toss the cubed potatoes with about 2 tablespoons of the pesto mix until evenly coated, then spread in a single layer on one side of the prepared pan. Roast for 15–20 minutes, depending on potato size, until starting to soften.
While potatoes begin roasting, add the chicken chunks, broccoli florets, and sliced carrots to the remaining pesto mix in the bowl and stir until everything is evenly coated.
Remove the pan from the oven and push the potatoes to one side, keeping a single layer. Add the coated chicken and vegetables to the other side and spread in a single layer.
Bake for 15 minutes, then remove from oven and evenly sprinkle with the grated Parmesan. Return to oven and bake 5 more minutes, or until chicken is cooked through and vegetables are tender. Optional: broil briefly to brown the cheese and vegetables.