Prepare the barbecue for medium heat with a direct and indirect zone. Add some beech wood smoking chips for an extra smoky flavour.
Brush the pointed peppers with olive oil and grill them for 3-4 min. cut-side down over direct heat until lightly charred. Flip, grill for another 2 min., and move to the indirect zone.
Open the can of tuna and place it on the indirect zone of the barbecue. Close the lid and heat for 5-8 min. so that the tuna absorbs a subtle smoky flavour.
Meanwhile, heat a splash of olive oil in a cast iron skillet or BBQ pan over medium heat. Add the anchovies and cook until they have melted into the oil.
Add a little oil from the tuna can, followed by the red onion, garlic, and shiitake mushrooms. Sauté for 3-4 min. until soft.
Add the tomatoes, thyme, and rosemary and cook for another 3-4 min. Stir in the capers and olives and heat briefly.
Divide the mixture over the grilled pointed peppers. Distribute the warm, smoky tuna in coarse pieces over them and drizzle with a little oil from the can.
Finish with lemon zest, basil, and freshly ground black pepper.
