Toss the diced chicken with the olive oil, garlic powder, onion powder, paprika, salt and pepper until lightly coated.
Heat a large nonstick skillet over medium heat. Add the chicken in a single layer and cook for about 6–8 minutes, turning occasionally, until browned and the center of a larger piece reaches 165°F (74°C). Cook in batches if needed. Let watery juices cook off, then transfer the chicken to a clean bowl.
Wipe loose browned seasoning bits from the skillet. Warm stiff tortillas briefly in the dry pan until pliable. Spread 2 tablespoons garlic sauce over the lower center of each tortilla, leaving the edges bare. Divide the chicken among them and add ¼ cup cheddar and 2 tablespoons mozzarella to each.
Fold the left and right sides over the filling. Bring the bottom edge up over the chicken and roll away from you, tucking the filling inside. Rest each wrap with its final overlap seam underneath.
Set the wiped nonstick skillet over medium-low heat. Place two wraps seam-side down in the dry pan, leaving room to turn them. Toast for about 2–3 minutes, until the underside has golden patches and the seam holds together.
Turn the wraps with a spatula and toast the other side for about 2–3 minutes, until golden and the cheese is melted. Lower the heat if they brown too quickly. Repeat with the remaining two wraps.
Serve the whole wraps while the tortillas are crisp. Add optional parsley if you like. Extra garlic sauce for dipping is optional and is separate from the ½ cup used in the filling.