Spread pesto mixture on both, top and bottom buns, making sure to cover the entire surface.
Place a couple slices of mortadella on each bottom bun followed by sweet capocollo and then prosciutto.
Top with red onions and roasted red peppers; season with freshly ground black pepper. Place tomato slices and Provolone cheese.
Place top buns and press down gently to compress. Brush the tops with olive oil.
Using a panini pan or press, place the buns, oiled side down, put the press on top or close the lid and grill on medium-high heat for 1 to 1:30 minutes.
When just about done, brush the top with the remaining oil, carefully flip the sandwiches, put the press on top and grill until golden as well.
Notes
Footnote: If it comes in long, thin strips, simply fold it in half.