Top with bun. Cut burgers in halves and serve immediately. Season with salt and freshly ground pepper. Grill chicken breasts over coals, or under broiler until firm to touch, about 8-10 minutes per side, less if thin, more if large. Transfer to plate and let cool about 10 minutes.
Slice rolls in half.
Sprinkle bottom of each roll with vinaigrette. Divide arugula among four rolls. Layer 2 slices of prosciutto and 2 slices of provolone on each roll. Drizzle more vinaigrette over chicken. Place top of roll on and press down gently so sandwich stays together.