For the tomatoes: Adjust oven rack to the lowest position and preheat oven to 300ºF.
Cut tomatoes into quarters from top to bottom. Working with one tomato piece at a time, start at the base and run knife parallel to cutting board to remove all of the seeds. To save time, you can angle the knife downward as you reach the hard white stem and remove it simultaneously.
Place the now-seedless tomatoes on a parchment-lined sheet tray. Drizzle with olive oil and sprinkle with salt to taste. Roast until tomatoes are wilted and slightly brown around the edges, about 30 minutes.
For the balsamic reduction: Heat balsamic vinegar in a small saucepan over medium heat. Cook until vinegar is bubbling and a rubber spatula leaves a trail that fills in after 1 second. Immediately transfer to a bowl. Vinegar will continue to thicken as it cools.
For the sandwich: Preheat a pan over medium heat. Nonstick works best, but you can use a well-seasoned carbon steel or cast iron pan.
Take 2 adjacent slices of bread and open them like a book. Sit them on a preparation surface. Distribute the cheese on one slice of bread.
Add butter to the pan. Tilt and swirl the pan to distribute.
Place both slices of bread on top of the melted butter. Move them around so that all of the butter is absorbed and evenly distributed on the bread.
Cook until both slices are golden brown, rotating bread as necessary to achieve even browning. Remove cheese-less slice and set on cutting board. Carefully flip other slice face-down in the pan until cheese is fully melted, about 30 seconds. Remove from pan.
Place tomatoes on the slice with the cheese and drizzle with as much of the balsamic reduction as you like. Top with basil leaves. Flip other slice of bread on top to assemble the sandwich.
Completed sandwiches can be held in a 225ºF oven for up to 30 minutes if necessary.