2 firm, ripe tomatoes (not overly ripe or they will fall apart when you grill them), cut in half around the middle
Kosher salt (or fleur de sel if you can get it)
Freshly ground black pepper
4 basil leaves, thinly sliced (roll them up like a cigar and then slice them to make nice thin slices)
Standing over a sink, use your fingers to gently dislodge and remove the watery pulp and seeds.
(Although the seeds and pulp contain a lot of flavor, when you grill the tomatoes they are going to be lost anyway.)
Preheat your grill on high heat for direct grilling.
Use a grill basketor fine grill grate if you can, it will make it much easier to work with the tomatoes (or any other vegetables you grill).
Season the tomatoes with salt and pepper. Brush the cut side of the tomatoes with olive oil. Brush the grill grate or grill pan with olive oil.
Place the tomatoes, cut side down on the grill surface. Cover the grill and let cook for about 4 minutes. (Check after 2 minutes.)
Use a metal spatula to lift tomatoes off of the grill for serving.
3 Place on a serving dish, cut side up. Drizzle a little more olive oil on the tomatoes.
Sprinkle with just a little more salt and pepper. Sprinkle with thinly sliced basil.