Cut the Hawaiian sweet rolls in half. Set aside.
In a medium bowl, combine the vinegar, dried mustard and oregano, salt, and pepper to taste. Whisk to mix all the ingredients. Pour in the olive oil and continue to whisk until it is combined. Set aside.
Spread the mayonnaise and mustard on the bottom half and only mayonnaise on the top half. Spread the Italian pickled vegetables over the mustard-covered bottom half.
Start layering your different Italian meats on the pickled vegetables. Then, add a layer of provolone cheese.
Sprinkle a handful of shredded iceberg lettuce to cover the cheese layer, top with tomato rings, and drizzle some of the sauce oil-vinegar sauce. Finally, cover with the top half of the bread roll to complete the mini godfather panini.
Pour the melted butter into a bowl and add the Italian seasoning, stir to combine. Pop it in the microwave to warm for 1- 2minute.
Use a pastry brush to spread the butter mixture on the top of the bread rolls.
Turn on the panini machine and let it heat up (until the green light comes on). Place the mini godfather panini on the press and cover it with the grill press for 3-5 minutes, or it turns a golden-brown color and has appetizing brown grill marks. Serve with spicy marinara dip.