Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Brush olive oil over both sides of each eggplant slice. Arrange eggplant slices onto a baking sheet.
Cook eggplant slices under the broiler until tender and lightly browned, about 5 minutes per side.
Toast split rolls under the broiler, cut sides up, until lightly browned, 2 to 3 minutes.
Stir yogurt, cheese, and garlic together in a small bowl until smooth; spread onto the rolls. Divide eggplant slices between the rolls; add tomato slices and basil to complete sandwiches.
