In the bowl of a food processor combine the basil, garlic, salt, and black pepper. Pulse to finely chop. Add the oil and process to blend. Transfer to a bowl.
Lightly brush the steaks with oil and season evenly with the salt and black pepper. Let the steaks stand at room temperature for 15 to 20 minutes.
Prepare the grill for indirect and direct cooking over high heat (450° to 550°F).
On a stovetop or side burner, preheat a large cast-iron griddle or skillet over medium-high heat. Drizzle the tomatoes with the oil and season with the salt and pepper. Add the tomatoes on the vine, and any loose tomatoes to the skillet and cook over medium-high heat until the tomatoes are begin to char and blister without losing their shape, 3 to 5 minutes. Remove from the heat and transfer to a plate.
Brush the cooking grates clean. Grill the steaks over direct high heat, with the lid closed, until an instant-read thermometer inserted into the center of the steak registers 125°F (or your preferred doneness), 6 to 8 minutes for medium rare, turning once or twice. Transfer the steaks to a platter and rest at room temperature indoors, for 3 to 5 minutes.
Serve the steaks with the tomatoes on the vine and basil coulis.