Preheat broiler.

Roast bell pepper until skin is blackened.

Place roasted pepper in a bowl, cover with plastic wrap, and let steam to loosen skin.

Peel the skin off the bell pepper and cut into 1-inch-wide strips.

Shave Parmigiano-Reggiano with a vegetable peeler to measure about 1/2 cup.

Place chicken between 2 sheets of dampened plastic wrap.

Pound chicken to about 1/4 inch thick with a rolling pin or meat pounder.

Arrange chicken on a lightly oiled rack of a broiler pan.

In a small bowl, whisk together balsamic vinegar, 1 tablespoon olive oil, black pepper, and salt to taste.

Brush chicken with vinaigrette.

Broil 2 to 3 inches from heat for 5 minutes.

Turn chicken over and brush with vinaigrette.

Discard any remaining vinaigrette and broil chicken until just cooked through, about 5 minutes more.

Transfer chicken to a plate and keep warm, loosely covered with foil.

Cut four 1/2-inch-thick slices from the middle of the crusty loaf of bread.

Lightly brush both sides of each slice with the remaining 3 tablespoons of olive oil.

Toast bread slices on the broiler pan under the broiler.

Halve the garlic clove crosswise and rub over 1 side of each slice of toast.

Assemble sandwiches with chicken, peppers, cheese shavings, and basil.

Season with salt and freshly ground pepper.