Preheat a gas grill to high (450°F to 500°F) on 1 side, or push hot coals to 1 side of a charcoal grill. To butterfly the steak, place steak on a cutting board with 1 long side facing you. Trim short edges as needed to form a rough rectangle. Press 1 hand flat on steak to steady it. Holding knife parallel to cutting board and starting on one short side, slice horizontally to within 3/4 inch of the opposite side. Open cut steak like a book.

Sprinkle cut sides of steak evenly with garlic, 1 1/2 teaspoons of the salt, and 1/2 teaspoon of the black pepper. Top with basil, leaving a 1-inch border on all sides. Arrange bell peppers and cherry bell peppers evenly over basil, leaving a 1-inch border on all sides. Overlap cheese slices on top of peppers, leaving a 1-inch border on all sides. Roll up steak away from you into a tight cylinder.

Tie cylinder crosswise at about 1 1/2-inch intervals starting from center and working out toward the ends using kitchen twine. Rub with oil. Sprinkle with remaining 1 teaspoon salt and 1/2 teaspoon black pepper.

Place steak on oiled grates over lit side of grill. Cook, uncovered, turning occasionally, until grill marks appear all over, about 10 minutes. Transfer to unlit side. Grill, covered, turning occasionally, until a thermometer inserted into thickest portion of steak registers 130°F to 135°F (medium-rare), 30 to 40 minutes, or to desired degree of doneness. Transfer steak to a clean cutting board. Let rest 10 minutes. Remove and discard kitchen twine. Cut crosswise into 8 slices.