Lay the bread slices on a clean work surface, and spread 1 1/2 teaspoons of the pesto over one side of each slice of bread. Divide the cheese evenly among the slices. Cook's Note: depending on the size of your cheese slices, you may need to cut them in half so that you can have cheese on both sides. Divide the roast turkey slices evenly among 4 of the bread slices. Place the remaining cheese-topped slices on top of the turkey-topped slices to form 4 sandwiches. Brush the outside of each sandwich with some of the olive oil.
Preheat a grill pan over medium heat. When it is hot, add the sandwiches, in batches if necessary, and weight them with a sandwich press or another skillet (or other heavy object). Cook until the sandwiches are golden brown and crisp and the cheese has melted, 4 to 6 minutes per side. Remove the sandwiches, cut in half on the diagonal, and serve immediately, with a simple side salad of hearts of romaine and tomato, if desired.
Though we prefer the crisp, ridged exterior you get when these sandwiches are cooked in a grill pan, they can also be cooked in a saute pan or panini press; the cook time will vary slightly.