Toss the tomatoes with the olive oil, thyme, red pepper flakes and garlic in a bowl. Season with salt and pepper.
Fit a baking sheet with a wire rack and arrange the tomato halves cut-side up on the rack. Cook the tomatoes until shriveled and tender, 1 1/2 to 2 hours.
Serving suggestions: Roughly chop the tomatoes, transfer to a serving bowl, top with freshly grated Parmesan and serve with crostini. Use as a condiment for roasted fish or chicken or stir into store-bought tomato sauce to improve the flavor.