Preheat a barbecue or grill pan over medium-high heat. Brush the eggplant and mushrooms with olive oil and season with salt and pepper.
Grill the vegetables for 3–4 minutes per side until tender and lightly charred. In the final minute, brush both sides with char siu sauce and cook until glossy and caramelised. Set aside.
Lay the flatbreads on the barbecue or a hot dry pan for 1–2 mins each side until warm and lightly toasted. Scatter with mozzarella and let it melt slightly from the residual heat.
Top each flatbread with the charred eggplant and mushrooms. Sprinkle with coriander and spring onion.