Preheat a panini press according to manufacturer's instructions.
Grill red bell pepper slices in the panini press on high heat until soft, about 5 minutes. Transfer to a blender.
Place chopped tomato and garlic in the blender. Blend with bell pepper until a smooth sauce forms, about 2 minutes. Season sauce with oregano, salt, and black pepper.
Grill eggplant, zucchini, tomato, and onion slices in the panini press until softened, about 3 minutes per side.
Drizzle olive oil over bread slices; flip and spread with a generous amount of sauce. Place 2 slices bread, oil-side down, on the panini press; add 1 slice of mozzarella cheese. Top with eggplant mixture and second slice of mozzarella cheese. Place second slice of bread, oil-side up, on top.
Grill on medium-high until golden and cheese is melted, about 5 minutes.