Combine mayonnaise, basil, lemon juice, salt and pepper in a small bowl; refrigerate until needed.
Preheat the oven to 425 degrees F. Place the breadcrumbs on a wide plate. Season the fish with salt and pepper. Line a baking sheet with parchment paper; brush it with the oil. Set a piece of the fish on the crumbs; pressing them on until the fish is lightly coated. Set the fish on the baking sheet. Repeat with remaining fish.
Bake the fish for 8 minutes, and then carefully turn each piece over. Top each piece with an equal amount of shrimp and cheese. Bake for 4 to 5 minutes more, or until the fish is cooked and the cheese is melted.
Brush the cut side of each bun with the basil mayonnaise. Set a lettuce leaf and tomato slice or two on each bottom bun. Set on the fish and serve.