In a shallow bowl, combine panko, garlic powder, and 1 cup Parmesan. Into another shallow bowl, pour flour. In a third shallow bowl, beat eggs with 1 tablespoon water.
Pat chicken dry; season with salt and pepper. Dip into flour, shaking off excess, then into egg, then into panko mixture, gently pressing to adhere.
In a small bowl, mix mozzarella and remaining 1 cup Parmesan.
Heat broiler. In a large, heavy pot over medium heat, heat oil until a deep-fry or instant-read thermometer registers 350°; if it smokes a bit, that’s okay. Set a wire rack in a baking sheet.
Working one at a time, fry chicken until one side is golden brown, about 1 minute. Flip and continue to fry until golden brown on the other side and the sound of the frying chicken gets slightly louder—this is the moisture evaporating from the chicken—about 1 minute more. Transfer to prepared rack. Repeat with remaining chicken. Spoon some pesto over chicken. Sprinkle with cheese mixture.
Step Broil, watching carefully, until cheese is melted and brown spots form, about 2 minutes.
Divide chicken among plates. Top with basil. Serve with remaining marinara alongside.
