Prepare the Chicken: Preheat oven to 350°F. Place chicken breasts between 2 sheets of plastic wrap, and flatten to about 3/4-inch thickness, using a rolling pin or flat side of a meat mallet. Sprinkle chicken with 1 teaspoon each of the salt and pepper.
Whisk together eggs and milk in a shallow dish. Stir together panko, Italian seasoning, smoked paprika, and remaining 1 teaspoon salt and 1/2 teaspoon pepper in another shallow dish. Dip chicken, 1 piece at a time, in egg mixture, shaking off excess; dredge in panko mixture, coating well and pressing to adhere.
Melt 1 1/2 tablespoons of the butter with 1 1/2 tablespoons of the olive oil in a large skillet over medium. Cook 3 chicken breasts in skillet until golden, 3 to 5 minutes per side. Remove chicken from pan, and place on a rimmed baking sheet. Repeat with remaining butter, oil, and chicken. Bake in preheated oven until chicken is cooked through, about 10 minutes.
Prepare the Herbed Rice: Combine all ingredients in a bowl, and stir until butter is melted. Serve with chicken and Pan Gravy.