Bring 1/2 cup water to a boil in a small saucepan. Stir in 1/4 cup uncooked quinoa, cover, reduce heat to low, and simmer until water is absorbed, about 15–20 minutes. Remove from heat and let cool.
Preheat the oven to 400°F (200°C) and lightly grease a baking sheet with cooking spray.
Place the remaining 1/4 cup uncooked quinoa into a food processor or spice grinder and process until it becomes a fine flour. Transfer the quinoa flour to a shallow plate.
In a mixing bowl, combine the cooled cooked quinoa, 1/4 cup grated goat cheese, Italian seasoning, garlic powder, and salt and pepper to taste. Mix well, then spread this mixture onto a second shallow plate.
Place the egg white on a third shallow plate and lightly beat if needed to loosen.
Pat the chicken breasts dry. Dredge each breast first in the quinoa flour, pressing to coat evenly, then dip into the egg white, and finally press into the quinoa-cheese mixture to coat all sides thoroughly.
Arrange the coated chicken breasts on the prepared baking sheet and lightly spray the tops with cooking spray.
Bake until the crust is lightly golden and the chicken reaches an internal temperature of 165°F, about 20–25 minutes.
Switch the oven to high broil. Sprinkle the remaining 1/4 cup grated goat cheese over the chicken and broil until the cheese melts and lightly bubbles, about 1–2 minutes—watch closely to avoid burning.
Garnish the chicken with sliced roasted red peppers and sliced fresh basil before serving.