Preheat the oven to 400°F (200°C). Lightly spray a 9x12-inch baking pan with cooking spray and set aside.
In a small bowl, whisk together the grated Parmesan, paprika, flour, 1 teaspoon salt, 1/4 teaspoon pepper, and garlic powder until evenly combined.
Place the beaten eggs in a separate shallow bowl. Dip each chicken breast first in the eggs, then press into the Parmesan-flour mixture to fully coat. Arrange the coated chicken in the prepared baking pan.
In a large bowl, combine the chopped broccoli, chopped red and yellow peppers, and chopped baby red potatoes. Toss with 1 tablespoon olive oil and season with additional salt and pepper to taste.
Spread the seasoned vegetables around the chicken in the baking pan so everything is in a single layer as much as possible.
Bake for 40–45 minutes, or until the chicken is golden and reaches an internal temperature of 165°F (74°C) and the potatoes are tender.
Let the dish rest for a few minutes before serving.