Preheat the oven to 400°. In a saucepan of boiling water, blanch the tomatoes until the skins start to curl, 30 seconds; transfer to a plate and let cool. Peel the tomatoes. Halve them crosswise and squeeze out the seeds, then slice the tomatoes 1/2 inch thick. Sprinkle the tomatoes on both sides with the sugar and season with salt; transfer to a rack and let drain for 30 minutes.
Meanwhile, line a large baking sheet with parchment paper. On a lightly floured work surface, roll out the puff pastry 1/8 inch thick. Using a 6-inch round plate as a template, cut out four 6-inch rounds. Transfer the rounds to the prepared baking sheet and freeze for 5 minutes. Bake the rounds for 20 minutes, until golden brown and puffed.
Pat the tomato slices dry with paper towels. Sprinkle the pastry puffs with half of the grated cheese and arrange the sliced tomatoes in a circle in the center of each pastry round, leaving a 1/2-inch border. Dot the tomatoes evenly with the butter and sprinkle with the remaining cheese.
Bake the tarts for 10 minutes, then reduce the oven temperature to 350° and bake for 20 minutes longer, until the pastry is richly browned and the tomatoes have shrunken slightly. Let the tarts cool for at least 5 minutes, garnish with the basil and serve.