Preheat the oven to 400˚F (200˚C). Oil a 9-inch pie dish or similar sized baking dish.
Spread the bread cubes on a small baking sheet and bake until lightly toasted, about 5-8 minutes.
Cut off the tops of the tomatoes (remove about ¼ inch). Using a small serrated knife (a steak knife works great), cut around the inside of the tomato, leaving a ¼-inch border. Use a spoon to scoop the flesh and seeds into a strainer set over a bowl. Using kitchen scissors, finely chop the tomato flesh and seeds in the strainer. Let sit 5 minutes to drain.
Arrange the tomatoes, cut sides up, in the baking dish.
In a medium bowl combine the toasted bread cubes, chopped tomato flesh (discard the juice in the bowl or drink it!), Comté, garlic and basil. Season with salt and pepper.
Season the tomatoes with salt and pepper. Mound the stuffing into the cavities.
Bake until the tops are golden brown, about 18-20 minutes. Let cool slightly. Drizzle with the best olive oil you’ve got, and garnish with more basil.