Whisk the olive oil, vinegar, oregano, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper in a small bowl to blend.
Arrange the slices of bread on a flat work surface and spread the vinaigrette on one side of each slice. Divide the mozzarella equally among 4 of the bread slices. Top with the prosciutto and roasted red peppers, then place the remaining 4 slices of bread on top, vinaigrette-side down, to form 4 sandwiches.
Heat a grill pan over medium heat. Add the sandwiches and cook, pressing them occasionally with a large spatula or the bottom of a small heavy saucepan, until the bread is golden brown and the cheese has melted, about 4 minutes per side.