Preheat the barbecue for medium-high heat and lightly oil the grate.
Place the capsicum onto the preheated grate and barbecue until the skin is completely charred on all sides, 10 to 15 minutes. Place the capsicum into a plastic bag, seal the bag and let cool. When cooled remove the charred skin, slice the capsicum into quarters and remove the seeds. Thinly slice the capsicum and set aside.
Mix salt and pepper into the olive oil in a small bowl. Brush the smooth side of the mushrooms with the seasoned oil and barbecue, oiled side down, until the mushrooms show grate marks, about 3 minutes. Brush the gill sides of the mushrooms with more seasoned olive oil, flip the mushrooms and cook until the mushrooms are softened and juicy, about 3 more minutes. Set the mushrooms aside and keep warm.
Lightly cook the slices of onion until softened and lightly browned, about 2 minutes per side. Spread the rolls out onto the barbecue to toast, about 1 minute. Mix the mayonnaise and roasted garlic in a bowl.
Spread each roll with garlic mayonnaise and make each sandwich with 1 mushroom cap, sliced mozzarella cheese, 1 slice of tomato, 4 basil leaves, 1 slice of grilled onion and 1/4 the roasted capsicum slices.