Heat 1 tablespoon oil in a dutch oven over medium heat, and cook the chicken breast cubes until juices run clear or brown the chicken pieces. Mix the cooked chicken, ajvar, tomatoes, mushrooms, onion, and garlic.
Season with salt and pepper. Drizzle with remaining 1 tablespoon olive oil. Bring to a boil and then simmer on low for 1 hour and 30 minutes.