Preheat the oven to 250 degrees F. Line a large baking sheet with parchment paper. Cut each tomato in half and squeeze out and discard the seeds. Dice the tomato flesh and place it in a bowl. Mix in the onion, garlic, oil, vinegar, salt and pepper. Spread the tomato mixture onto the baking sheet, and then roast for 90 minutes, or until very tender and tastily aromatic.
Cool the mixture to room temperature. Stir in the basil and add additional salt and pepper, if needed. Spoon the relish into a serving bowl and set on a platter. Set the cheese beside the relish and serve.