Whisk 3 tablespoons of the olive oil, vinegar, basil, garlic, salt, and pepper in a small mixing bowl to blend. Set the vinaigrette aside.
Arrange the slices of bread on a flat work surface and, using a pastry or basting brush, brush one side of each slice with the vinaigrette, dividing it equally.
Divide the mozzarella equally among 4 of the bread slices. Top the mozzarella with the pepperoni slices, then place the remaining 4 slices of bread on top of the mozzarella and pepperoni, brushed sides down, to form 4 sandwiches.
Brush the outsides of each sandwich lightly with some of the remaining tablespoon of olive oil.
Heat a large skillet or grill pan over medium heat. When the skillet is hot, add the sandwiches and cook until the bread is golden brown and the cheese is melted, pressing occasionally with a large metal spatula or the bottom of a small heavy saucepan, 4 to 5 minutes per side. Serve hot.
Yield: 4 sandwiches