Whisk 1/4 cup olive oil, vinegar, oregano, garlic, salt, and pepper in a small bowl to blend. Arrange the slices of bread on a flat work surface and, using a brush, divide the vinaigrette equally among 1 side of each slice. Divide the mozzarella equally among the bread slices. Top 4 of the slices of bread equally with the prosciutto and soppressata and then place the remaining 4 slices on top.

Brush the outsides of each sandwich with some of the remaining tablespoon of olive oil. Heat a large skillet or grill pan over medium heat. Add the sandwiches and cook until the bread is golden brown and the cheese is melted, pressing occasionally to compact with a large spatula or the bottom of a heavy small saucepan, about 4 to 5 minutes per side.