Whisk 6 tablespoons of the 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 provolone equally among the bread slices.
Top 6 of the slices of bread equally with the sliced artichoke hearts and sliced genoa salami and then place the remaining 6 slices on top.
Brush the outsides of each sandwich with some of the remaining 2 tablespoons of olive oil.
Heat a grill, 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.
Remove the sandwiches from the grill, cut in half and serve.