Beat eggs and milk in a shallow dish. Mix flour, salt and cayenne pepper in another shallow dish. Dredge catfish fillets in flour mixture, then in egg mixture and again in flour. Set aside. Heat oil in a deep fryer or a large, deep skillet to 360° F. Fry fillets, 2 at a time, for 2 to 3 minutes on each side, or until golden brown and fish flakes easily when tested with a fork.
Place on paper towels to drain. Prepare the sauce. Heat 1 tablespoon of the melted butter in a large skillet over medium heat. Add shrimp, and sauté until light pink on both sides; do not overcook. Stir in garlic, ham strips and vermouth.
Mix in cream, half of the green onions, lemon juice, salt and cayenne pepper; cook for 1 to 2 minutes to reduce cream. Remove from the heat, and whisk in remaining 2 tablespoons of melted butter.
Place a catfish fillet on each serving plate, and top with 3 shrimp pieces. Arrange ham strips on top. Spoon sauce onto catfish, and sprinkle with remaining green onions. Garnish with lemon wedges and parsley sprigs, and serve.