Preheat the oven to 325 degrees and position 2 racks just above and just below the middle of the oven. Working over a medium bowl, pry the seeds and pulp out of the tomatoes and discard. Pour 1/4 cup of the olive oil onto each of 2 very sturdy, rimmed, light-colored baking sheets. Arrange the tomatoes cut side down on the baking sheets and scatter the garlic and thyme all around. Make a tiny slit on each tomato.

Bake the tomatoes for about 45 minutes, until the skins begin to wrinkle. Shift the pans from top to bottom halfway through. Carefully pinch off the skins. Flip each tomato and bake until the surface looks dry, about 1 hour. Flip the tomatoes again and continue baking until the surfaces look dry but the tomatoes are still slightly plump, about 2 hours longer. Season the tomatoes with salt and pepper and let cool completely.

Discard the thyme sprigs and peel the garlic cloves. Layer the tomatoes with the garlic in five 1/2-pint jars. Add enough olive oil to cover the tomatoes by at least 1 inch. Slide the blade of a knife along the side of each jar to release any air bubbles. Seal the jars and refrigerate for up to 2 months or freeze for up to 6 months.