Special equipment: Plancha (flat top grill), bricks wrapped in foil
Prepare a grill for direct and indirect grilling over medium heat. Heat the plancha, or a cast-iron griddle, over the direct heat.
Spread the mayonnaise on the bottom of the rolls. Top with half the Manchego and ham. Then, top with the Spanish Spice Rubbed Pork Butt and arugula. Top with the remaining half the Manchego and ham. Spread the mustard on the top half of the rolls. Place the tops on the sandwich and brush with the Piquillo Pepper Butter.
Place the sandwiches butter-side down on the griddle. Place the bricks on top of the sandwiches and cook until the bread is crisp and golden brown, about 5 minutes. Watch carefully: if the bread browns too quickly, move the plancha to indirect heat and cover the grill to melt the Manchego. Flip the sandwiches over and place the bricks back on top. Continue cooking until the bottom of the sandwiches are crisp and the Manchego is melted, about 3 minutes.
Blend the butter, piquillo peppers, garlic, thyme, paprika, salt and pepper in a food processor until smooth. Scrape the butter into a bowl, cover and refrigerate for at least 1 hour and up to 24 hours before using. Soften the butter slightly before using.