Mix bread crumbs, roasted peppers, 2 tbsp oil, capers, parsley, garlic, red pepper flakes, and 1/2 tsp salt in a bowl.
Season steak with salt and pepper. Butterfly steak. Layer ham, cheese, and seasoned crumbs over the steak.
Starting with short side, roll and tie steak. Season with salt and pepper, brush with 2 tbsp oil, and refrigerate while heating grill. Steak can be wrapped and refrigerated for 1 day.
Grill steak over fire until browned on all sides, about 12 minutes. Move steak to cool side of grill and brush with 2 tbsp oil. Cover grill and cook until instant-read thermometer registers 120 degrees, 20 to 30 minutes, rotating and brushing steak every 10 minutes with 1 tbsp oil.
Transfer steak to cutting board, tent with foil, and let rest 10 minutes. Remove twine and slice steak crosswise into 1/2" thick slices.