Cut the top off the tomatoes and clean out the seeds and ribs. Season the center with salt and pepper. Cut the cheese into circles to fit inside the tomatoes.
On the grill: Prepare the grill to a medium heat and brush slightly with oil. Place tomatoes on the grill and once they begin to char and the cheese softens, transfer them to a baking pan and set aside; keep warm. Sprinkle with chopped dill before serving.
In the broiler: Place tomatoes under the broiler. Once the tomatoes begin charring and the cheese softens, remove and keep warm until ready to serve. Sprinkle with chopped dill before serving.