Preheat oven to broil. Halve each pepper, discarding stems and seeds. Place peppers, cut side down, on an old baking sheet or one lined with foil.
Cut eggplant in half lengthwise and score with a knife, drizzle it with about 2 tablespoons olive oil and a little salt and place it on a second baking sheet.
Place one of the oven racks roughly 3 to 4 inches below the heat; place peppers on this rack. The eggplant should sit on a lower rack.
Broil the peppers and eggplant, turning the peppers occasionally until they are well roasted on all sides, roughly 15 to 20 minutes. The eggplant may be done first; if so, remove it and set it aside to cool. To test eggplant for doneness press on it with a fork to make sure it's soft all the way through. I use an an outdoor grill to cook the peppers and eggplant, no oil. I make sure most of the skin on the peppers are blackened.
When you remove the peppers from the oven, place them in a bowl, sprinkle them with a little water, and cover with a clean dishcloth. Leave it for about 10 minutes. This step steams and lifts the skins, making it easy to peel the peppers once they've cooled. (Another method to loosen skins is to place the peppers in a paper sack, fold over to close and leave it for 15 minutes).
While waiting for the red pepper skin to lift. Use a large spoon to remove the pulp of the eggplant, leaving the skin behind. Discard the thickest seeds, but don't get nit-picky. Put eggplant in a food processor or a blender (I use a stick blender in a deep bowl) with olive oil and smashed and chopped garlic cloves. Pulse the eggplant a few times so that it's chopped with the other ingredients but not totally pureed. Put this in a bowl and set aside.
Once peppers are cool enough to handle, peel them with your fingers or a knife. Add peppers to the food processor or blender and pulse 5 to 8 times to chop coarsely. Add the eggplant mixture to the chopped peppers, season to taste with salt, pepper, vinegar and dried chili flakes, pulse 3 or 4 times to mix.
If you wish, garnish it with more finely diced garlic and a few scattered basil leaves. Serve ajvar with toasted bread slices drizzled with olive oil and sprinkled with salt.