Slice the tomato and mozzarella, split the rolls, and pat the chicken dry.
Season chicken with salt and pepper. Heat olive oil in a skillet over medium-high.
Sear chicken 3–4 minutes per side until browned and cooked through. Rest 2 minutes, then slice if thick.
Reduce heat to medium. Spread butter on cut sides of the rolls and toast cut-side down until golden.
Spread pesto and a thin layer of mayo on toasted rolls. Layer chicken, mozzarella, tomato, and arugula. Sprinkle Parmesan; drizzle balsamic if using.
Return sandwiches to the skillet. Press lightly and toast 1–2 minutes per side until cheese softens and bread is crisp.