Place the halved chicken breasts in a mixing bowl and pour the balsamic dressing over them; stir to coat thoroughly, cover with plastic wrap, and refrigerate for 1 hour (or up to overnight).
Preheat the oven to 400°F (200°C). Heat an oven-safe sauté pan over medium-high heat and add the olive oil.
When the oil is hot and shimmering, remove the chicken from the marinade, season both sides with salt and pepper, and sear in the pan for about 3 minutes per side, until golden brown.
Transfer the pan to the preheated oven and bake for about 3–6 minutes per side, or until the internal temperature reaches 165°F (74°C) and the chicken is cooked through.
Set the oven broiler to high. Top each chicken piece with sliced fresh mozzarella and broil 2–4 minutes, watching carefully, until the cheese is fully melted and lightly bubbling.
Remove the pan from the oven and let the chicken rest 5–10 minutes on a cutting board before serving; garnish with fresh basil leaves if using.