Using a sharp paring knife, cut out cores from tomatoes; cut a small X on the other end. Place half of tomatoes in boiling water about 20 seconds; remove with a slotted spoon and plunge into ice water. Repeat with remaining tomatoes.
When cool, remove the skin from each tomato, cut in half, and squeeze out juices and seeds. Gently toss tomatoes in a large bowl with olive oil, garlic, thyme, salt, pepper, and sugar.
Arrange tomatoes, cut side down, on baking sheet. Bake until very soft, about 5 hours.
To store: Pack tomatoes in an airtight container, adding olive oil to cover. Store refrigerated up to 3 weeks.