How to Make It
In a mini-processor, blend the mayonnaise, Parmesan, anchovies, garlic, mustard, lemon juice and 1/4 teaspoon pepper; alternatively, combine the ingredients in a bowl.
Brush a grill pan or a cast-iron skillet with 1 teaspoon of the olive oil and heat. brush the chicken breasts and tenders on both sides with the remaining 2 teaspoons oil and season with salt and pepper. Pan-grill the breasts over high heat, turning once, until browned and cooked through, about 3 minutes per side. Let rest for 5 minutes.
Add the tenders to the pan and grill, turning once, until browned and cooked through, about 2 minutes per side. Cut the tenders in half lengthwise. Slice the breasts diagonally 1/2 inch thick.
Spread the Caesar mayonnaise on the top and bottom half of each roll. Arrange a sliced chicken breast and tender on the bottom half of each roll, cover with arugula and close the sandwiches.