Heat a gas grill to medium high. Put the arugula in a large bowl, cover with a damp paper towel, and refrigerate. In a blender, combine the lemon juice and zest, anchovies, garlic, and peppercorns with 6 tablespoons of the oil, 1 tablespoon of the mustard, and 1/2 teaspoon salt. Blend thoroughly until most of the peppercorns are well broken up and the dressing is emulsified. Add 2 tablespoons of the grated Parmigiano and blend to incorporate. Leave the dressing in the blender.

In a medium bowl, toss the chicken with the remaining 1 tablespoon mustard and 3/4 teaspoon salt. Brush the bread with the remaining 2 tablespoons oil and season with salt. Grill the chicken, covered, until golden grill marks form, 5 to 6 minutes (for even cooking, rotate the chicken 90 degrees halfway through cooking each side). Flip the breasts and continue to cook, covered, until golden grill marks form and the chicken is cooked through, 5 to 6 minutes. Let rest on a cutting board for 5 minutes. Grill the bread until dark around the edges and golden brown in the center, 1 to 2 minutes per side. Slice the chicken thinly on an angle. Cut each bread slice into 10 cubes. Pulse the dressing in the blender; add a little of it to the arugula, just to coat.

Divide the arugula among four plates and sprinkle with a little Parmigiano. Toss the chicken with the remaining dressing and arrange over the arugula. Sprinkle with the remaining Parmigiano and arrange the croutons and tomatoes (if using) around the salad.