To make the pesto oil, pulse the basil and garlic in a food processor or blender until finely chopped. With the machine running, add the olive oil in a steady stream and process until well blended. Season with the salt and pepper. Transfer to a bowl and cover tightly.
To make the napoleons, preheat the oven to 450°F.
Cut the eggplant crosswise into 12 slices about 1/4 inch thick. Mix 1/4 cup of the olive oil and the balsamic vinegar in a large bowl. Add the eggplant slices and toss to coat. Season with the salt and pepper.
Place the eggplant on a baking sheet in one layer and bake until soft, turning once, about 15 minutes. Let cool.
In a small bowl, combine the cheese, basil, and the remaining 1 tablespoon of olive oil.
Slice each tomato crosswise into 4 slices about 1/3 inch thick. Place 1 eggplant slice on a salad plate. Top with about 1 tablespoon of the cheese mixture, then a tomato slice. Drizzle 1 teaspoon of pesto oil over the tomato, then top with another tablespoon of cheese. Repeat with an other eggplant slice, 1 tablespoon cheese, a tomato slice, and 1 teaspoon pesto oil. Finish with a final eggplant slice, another tablespoon of cheese, and a drizzle of pesto oil. Repeat with the remaining ingredients.
Serve at room temperature, with pesto oil drizzled around each napoleon.
Yield: 4 servings