Makes about 1 quart

3 pounds tomatoes (beefsteak or plum), cored and halved
2 medium onions, halved and sliced 1/4 inch thick
2 carrots, sliced into 1/4-inch rounds
6 cloves garlic, unpeeled
1/2 teaspoon dried thyme
½ cup plus 3 tablespoons olive oil
Coarse salt and ground pepper

Preheat oven to 425 degrees. Use a sharp paring knife to core the tomatoes. Cut tomatoes in half; transfer to one large (or two smaller) rimmed baking sheet; add onion, carrots, garlic, and thyme.

Toss tomato mixture with oil; season generously with salt and pepper. Spread in a single layer (turn tomatoes cut side down). Roast until tender, 45 minutes to 1 hour. If vegetables begin to brown too quickly, push them toward the center of the sheet.

Using tongs or your fingers, peel off tomato skins; discard. Transfer (including juices) to a blender. Add the onion to the food processor. Peel the garlic cloves and drop them into the food processor with the tomatoes. Pulse several times, until desired consistency is reached. Add basil, balsamic vinegar, and the remaining 3 tablespoons of olive oil. Pulse to combine.

Season with salt and pepper. Let cool completely; transfer to an airtight container. Refrigerate up to 1 week, or freeze up to 3 months.