Place the tomatoes in a medium bowl. Sprinkle 1 teaspoon salt over the tomatoes and stir until coated. Set aside for 1 hour.
Preheat a grill for cooking over medium-high heat. Brush the grill grates with the vegetable oil. Sprinkle the steaks generously with salt and pepper on all sides. Place the steaks on the grill and cook, flipping only once, about 4 minutes per side or until desired degree of doneness is reached. Set aside to rest for 10 minutes.
Prepare the Texas toast on the grill, grilling about 2 minutes more per side than the package instructions call for to make the bread extra-crispy. Cut the toast into bite-sized pieces and set aside.
Pour off any juice that has collected from the tomatoes into a large bowl for the vinaigrette. Whisk in the vinegar and mustard. While continuously whisking, slowly pour in the olive oil until the mixture is throughly combined and emulsified. Add the tomatoes to the vinaigrette.
Cut the steaks into 1-inch cubes, discarding any large pieces of fat. Add the cubes of steak and toast to the tomatoes and vinaigrette and tear 5 basil leaves into the bowl. Toss gently to combine.
Pour the salad onto a serving platter and tear the remaining basil leaves over the top. Serve immediately.