Make the salsa verde: Combine the parsley, olives, cocktail onions, capers, cornichons and anchovy in a blender. Add the olive oil, mayonnaise, mustard and a splash of water; blend until smooth. (Do not blend too long or the parsley will turn brown.) Set aside.
Prepare the sliders: Cut the chicken breast horizontally into thirds to make 3 equally thin cutlets. Restack the chicken pieces and cut them in half crosswise to make 6 pieces. Season with salt and pepper.
Lightly beat the eggs in a shallow bowl and season with salt and pepper. Place the flour in another shallow bowl and season with salt and pepper. Combine the panko, parmesan, pecorino, oregano, sage, thyme, red pepper flakes and 1/4 teaspoon salt in a third shallow bowl. Dredge the seasoned chicken in the flour, then dip in the egg wash and coat with the panko mixture, pressing firmly to adhere.
Heat a 12-inch nonstick skillet over medium-low heat. When hot, add the vegetable oil to evenly coat the skillet. Add the chicken and cook until golden brown, about 3 minutes per side. Remove and drain on paper towels. Season with salt.
Smear salsa verde on both sides of the toasted buns, sandwich with the chicken and top with sliced tomatoes and arugula.