Preheat oven to 300°F (150°C). Line a rimmed baking sheet with parchment paper. Tear each tomato in half and press out and discard the seeds. Arrange tomatoes on prepared baking sheet in one layer, season with salt, and drizzle with olive oil. Cook until excess juices have evaporated and tomatoes look slightly dry on the exterior but still moist within, about 1 hour.

Transfer oven-roasted tomatoes to a work surface and chop finely.

Transfer chopped tomatoes to a large mixing bowl. Add 1/4 cup olive oil and basil, and stir well. Season with salt and add red wine vinegar, about 1 teaspoon at a time, until tomatoes are very lightly tart. Stir in a pinch of sugar to enhance tomatoes' natural sweetness; add more sugar sparingly to taste, if desired.

Rub top sides of toasts with garlic. Drizzle toasts with olive oil and season lightly with salt. Spoon tomatoes on top and serve.