Preheat a griddle or cast iron skillet. Spread each side of split roll with pesto or tapenade. Layer some turkey and cheese on one half of roll and top with other half.
Add a couple of teaspoonfuls of butter to skillet and add roll. Top with aluminum foil, put heavy weight on top and cook the sandwich, weighted down, slowly for 4 to 5 minutes. Turn sandwich over, add a couple more teaspoonfuls of butter to skillet and cook weighted down until cheese has melted--another minute or so. Serve hot and eat with knife and fork. Serve with Fennel and Roasted Pepper Salad below.