Whisk the olive oil, vinegar, oregano, garlic, salt, and pepper together in a small bowl to blend.
Arrange the slices of bread on a flat work surface, and using a brush, spread the vinaigrette evenly over one side of each slice. Divide the mozzarella equally among the bread slices. Top 4 of the bread slices with the prosciutto and red peppers, and then place the remaining 4 slices of bread on the top, vinaigrette-side down, to form 4 sandwiches.
Heat a large skillet or 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. Serve immediately, with a side salad of grape tomatoes, roasted red pepper and oregano with balsamic vinaigrette, if desired.