Special equipment: a meat mallet, butcher's twine and an instant-read thermometer
For the flank steak: Set the flank steak on a cutting board. Starting at the thin, narrow end of the steak, slice it in half horizontally to within 2 inches of the opposite wide end. Open it up like a book so you have one long piece. Pound the steak with a mallet to an even thickness of about 1/2 inch.
Whisk together the oil, vinegar, honey, red pepper, garlic and 1/2 teaspoon each salt and pepper in a medium bowl. Combine the marinade with the meat in a large resealable plastic bag and flip over several times to make sure the meat is well coated. Marinate in the refrigerator for 2 hours, flipping several times.
For the herb paste and sauce: Combine the basil, parsley, red pepper, garlic and 1 teaspoon salt in a food processor and pulse until finely chopped. With the motor running, slowly pour in 1/4 cup of the oil. Remove 2 tablespoons of the herb paste from the processor and mix with the remaining 1/4 cup oil in a small bowl; reserve this sauce for finishing.
Prepare an outdoor grill for direct and indirect grilling. (If your grill has a thermometer, aim to keep the temperature at about 350 degrees F throughout grilling.)
Remove the steak from the marinade and pat dry. Spread it flat on a work surface. Cover the steak with the remaining herb paste, leaving a 1-inch border along the edges. Layer with the ham, mozzarella and salami, then sprinkle with the mushrooms and roasted peppers. Arrange the hot peppers in a line along the edge of one of the short ends (when you roll up the steak they will be in the middle). Roll up the steak like a jelly roll, starting at the short end with the hot peppers. Tie the roll closed around the circumference with butcher's twine at 2-inch intervals. Tie one more piece of twine lengthwise around the meat.
Put the steak on the direct heat side of the grill and cook, turning, to brown all 4 sides, about 2 minutes per side. Move the steak to the indirect heat side of the grill, cover and cook, turning occasionally, until the very center reads 115 degrees F on an instant-read thermometer for a steak that is well done on the outside and medium rare in the center, 25 to 30 minutes more. (Test frequently; the time will vary based on the temperature of your grill.) Let the steak rest 5 to 6 minutes.
Untie the steak and cut it into 6 thick slices. Serve with the reserved herb sauce for drizzling.
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