Combine egg and egg white in a shallow bowl, stirring with a whisk. Combine breadcrumbs, 1/2 teaspoon salt, and 1/4 teaspoon pepper in another shallow bowl. Pat fillets dry. Dip fillets in egg, letting excess egg drip off. Lightly dredge fillets in breadcrumb mixture.
Heat 2 teaspoons oil in a large nonstick skillet over medium heat. Add 2 fillets; cook 2 minutes on each side or until fish flakes easily when tested with a fork. Transfer fillets to a serving platter. Repeat procedure with 1 teaspoon olive oil and remaining fillets.
Combine greens, tomato, and onion in a bowl. Combine remaining 1 tablespoon olive oil, remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, lemon juice, vinegar, and mustard, stirring well with a whisk. Pour dressing over greens; toss to coat. Top fillets with greens mixture, and sprinkle with parsley.