Preheat oven to 425°F. Set garlic head on its side on a cutting board. Using a sharp knife, cut about a ½-inch portion off the top part of the head. You should see most of the individual cloves revealed; if not, cut a little more off. Remove the loose papery peels from outside the head but do not take all of the peel off. Set the garlic head, top side up, in a custard cup. Drizzle with ½ teaspoon oil. Cover with foil.
Bake for 35 to 40 minutes or until garlic cloves are just soft when pierced with a sharp knife. Keep the garlic covered in the custard cup. Let stand on a wire rack until cool enough to handle (20 to 30 minutes).
Meanwhile, line a large baking sheet with foil. Arrange pepper quarters on foil, skin sides up. Arrange onion slices on foil with peppers. Brush onion slices with about 1/2 tablespoon of the oil, making sure to coat tops completely. Roast peppers and onion alongside the garlic for 25 to 30 minutes or until pepper skins are well charred (skin should be at least half covered with large black blisters/bubbles).
Set pan on wire rack. Remove any outer rings of the onion slices that look overbrowned and dry (you may not see any). Grab the corners of the foil and wrap them up around the peppers and onion, sealing the foil to completely enclose the vegetables. Let the packet cool 20 to 30 minutes or until vegetables are cool enough to handle (don’t skimp on the cooling step – it helps loosen the skin from the peppers!).
Place tomatoes in a 2-quart baking dish; drizzle with 1 tablespoon of the oil; toss to coat. Roast, uncovered, for 20 minutes or until tomatoes are slightly shriveled and starting to brown. (I put the tomatoes on a lower rack in the oven and roast at the same time as the peppers and garlic.)
When peppers are cool enough to handle, use a small sharp knife to peel off the charred skin; discard skin. It should peel off easily*.
When garlic is cool enough to handle, hold garlic head on the bottom. Pinch bottom of head with your fingers to squeeze out the individual cloves. You may have to use a sharp knife to remove some of them.
Transfer garlic cloves to a blender. Add peeled peppers and the onion slices to blender. Add tomatoes and any cooking juices in the pan. Add remaining 1 1/2 tablespoons olive oil, 1 tablespoon of the balsamic vinegar, the salt, and black pepper. Cover and blend until very smooth. Add thyme; cover and blend until combined. Taste and add the additional 1 tablespoon vinegar if desired.
This sauce can be served at room temperature, chilled, or warmed. It tastes great either way. Heating the sauce will mellow out the tang from the vinegar. You can always add a splash more vinegar if you want a bit more zing after it’s heated.