Quick-roast and peel bell pepper and cut into 1-inch-wide strips. With a vegetable peeler shave enough Parmigiano-Reggiano to measure about 1/2 cup.
Put chicken between 2 sheets of dampened plastic wrap and with a rolling pin or flat side of a meat pounder lightly pound to about 1/4 inch thick. Arrange chicken on lightly oiled rack of a broiler pan.
In a small bowl whisk together vinegar, 1 tablespoon oil, pepper, and salt to taste.
Brush chicken with vinaigrette and broil 2 to 3 inches from heat 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 middle of loaf and lightly brush both sides of each slice with remaining 3 tablespoons oil. Toast bread slices on broiler pan under broiler. Halve garlic clove crosswise and rub over 1 side of each slice of toast.
Assemble sandwiches with chicken, peppers, cheese shavings, and basil and season with salt and freshly ground pepper.
