Poke holes in eggplants with a fork. Arrange eggplants, green bell peppers, and red bell peppers on a baking sheet and spray with cooking spray.
Bake in the preheated oven, turning peppers with tongs every 15 minutes, until tender, 40 to 45 minutes. Remove; cover immediately with a damp towel. Let steam as they cool, about 20 minutes.
Scoop eggplant flesh and juice into a food processor. Add garlic; puree until smooth. Scrape into a large bowl. Season with salt and pepper.
Hold a bell pepper over the bowl of the food processor; discard stem and seeds and allow juices to stream into the bowl. Peel skin away slowly and discard. Drop bell pepper into the food processor. Repeat with remaining bell peppers. Pulse until finely chopped.
Scrape the chopped peppers into the bowl with the eggplant puree. Season with salt and pepper. Stir in vinegar. Stir in oil. Cover with plastic wrap; refrigerate until flavors are well blended, 8 hours to overnight.