For the chicken: Line a baking sheet with waxed paper. Preheat the oven to 200 degrees F.
Sprinkle the chicken breasts with salt and pepper. Place the flour on a plate. Beat the eggs and Dijon mustard to blend in a shallow baking dish. Mix the panko with the herbs, 1 teaspoon black pepper and 1 teaspoon salt. Place 1 cup panko mixture in another shallow baking dish, adding remaining cup panko, or more, to dish as needed throughout process.
Working with 1 chicken breast at a time, dredge in the flour, shaking off the excess, then dip into the egg mixture, turning to coat evenly, then carefully coat with the panko, pressing gently to adhere to the chicken. Transfer the chicken to the prepared baking sheet.
Heat 4 tablespoons butter in a 12-inch cast-iron skillet over medium-high heat. Add 1 chicken breast to the skillet and cook until golden brown on both sides, 8 to 10 minutes total. Transfer to a wire rack-lined sheet tray and season with salt. Repeat with the remaining butter and chicken breasts, keeping the chicken warm in the oven while you make the remaining batches, if desired.
For the salad: Meanwhile, add the currants to a small saucepan and cover with the vinegar, sugar and salt. Bring to a boil, then turn off the heat and let sit until cool to the touch, about 10 minutes.
Chop the romaine into small pieces and toss with the currants and Parmesan. Dress with the olive oil and pickling liquid to taste.
For the sauce: Add the butter to a saucepan and heat over medium heat until the butter browns, then add the anchovies, capers, olives and shallots and toast until slightly crispy, about 2 minutes. Add the lemon juice and emulsify.
Place the chicken on a serving plate or platter and drizzle the sauce over the chicken. Serve the salad alongside.