Preheat oven to broil with rack in top third position. Line a large baking sheet with foil. Combine 3 tablespoons mayonnaise, 3 tablespoons yogurt, 1 tablespoon mustard and 1 tablespoon vinegar in a large bowl; stir well.
Add flaked tuna, chopped eggs, 2 tablespoons basil, 2 tablespoons onion and ⅓ cup olives to the mayonnaise mixture; gently stir until thoroughly combined.
Brush both sides of 4 bread slices with 2 tablespoons oil; place on the prepared baking sheet. Broil until toasted, 1 to 2 minutes. Remove from oven. Mound each slice with a generous ½ cup tuna mixture. Arrange sliced tomatoes on the sandwiches; sprinkle with ½ cup cheese (about 2 tablespoons each). Broil until the cheese is melted and bubbly, 3 to 4 minutes. Top with the remaining 2 tablespoons olives. Garnish with basil, if desired.
