01 - Preheat BBQ grill or grill pan to medium-high heat (375-400°F).
02 - Combine olive oil, lemon zest, garlic, salt, and pepper in shallow dish. Add barramundi fillets, turning to coat. Marinate for 10 minutes at room temperature.
03 - Brush cos lettuce leaves lightly with olive oil on both sides.
04 - Place lemon halves cut-side down and jalapeño halves on grill. Cook 3-4 minutes until charred and softened. Remove and let cool slightly. Finely chop jalapeños. Squeeze juice from grilled lemons.
05 - Whisk mayonnaise, grilled jalapeños, 2 tbsp grilled lemon juice, capers, dill, parsley, shallot, Dijon mustard, salt, and pepper in bowl until combined. Refrigerate until ready to use.
06 - Place marinated barramundi on grill. Cook 3-4 minutes per side until just cooked through and lightly charred. Fish should flake easily when tested with fork.
07 - Grill cos lettuce leaves 1-2 minutes per side until slightly wilted with smoky char marks.
08 - Place split buns cut-side down on grill for 30-60 seconds until lightly toasted and golden.
09 - Spread generous layer of grilled lemon jalapeño tartare on both cut sides of each bun. Layer bottom bun with grilled cos lettuce, barramundi fillet, tomato slices, and cucumber slices. Top with bun crown.
10 - Serve burgers immediately while fish is hot and lettuce remains slightly crisp.