Put the olives, red peppers, cucumber, garlic, onion and vinegar in a medium bowl; season lightly with salt and toss to combine. Set aside at room temperature.
Put 1 of the buns in a food processor and pulse to make coarse crumbs.
Put the mayonnaise, mustard and egg whites in a large bowl and use a fork to combine. Add the tuna and 1/2 cup of the breadcrumbs (save the rest for another use) and season with salt and pepper. Toss well to combine without breaking up the tuna too much. Divide into 4 patties, each about 1/2 inch thick and 4 inches across. Put on a wax-paper- or plastic-wrap-lined plate and chill for 20 minutes.
Heat the oil in a large nonstick skillet over medium heat. Add the patties to the skillet and cook until golden brown on both sides, about 3 minutes per side.
Spread 1 teaspoon sun-dried tomato tapenade on each bottom bun and top with a tuna burger. Top with 2 to 3 basil leaves and about 1/3 cup reserved olive-and-cucumber salad and cover with the top buns. Serve.
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