Heat broiler to high with rack set 6-inches from element. Arrange bread slices on a baking sheet, brush one side with oil, and season with salt and pepper. Broil bread until lightly toasted, about 1½ minutes each side.
Transfer four bread slices to a cutting board, oiled-side down, and spread with Dijon.
Divide cheese among remaining four bread slices on baking sheet, oiled side down. Return cheese-topped bread to oven and broil until cheese is melted, about 1 minute.
Divide turkey, lettuce, basil, peaches, and onion among four cheese-topped bread slices. Place remaining four bread slices, Dijon-side down, on top sandwiches and cut in half.