Prepare a fire in a charcoal grill or preheat a gas grill.In a bowl, combine the chopped tuna, onion and its juice, mayonnaise, salt and pepper. Shape into 4 firm patties about 3 inches in diameter and 1 1?2 inches thick. Cover and chill in the freezer for no more than 10 minutes.Meanwhile, make the lemon-chive mayonnaise: In a small bowl, stir together the mayonnaise, salt, pepper, lemon juice, chives, soy sauce and hot-pepper sauce. Set aside.Using a wide spatula, carefully place the tuna patties on the grill. Grill the patties, gently turning once, until lightly browned, about 5 minutes total for medium-rare, 8 minutes total for medium. At the same time, lightly toast the bread slices on the grill.