Drain artichokes, reserving marinade; thinly slice artichokes. Combine artichokes, reserved marinade, peppers, olives, onion, garlic, parsley, oregano, and cumin. Cover; chill 2 to 24 hours, tossing occasionally. If using focaccia, cut in half crosswise. Arrange marinated vegetables, meat, and cheese over bottom halves of rolls. Add tops of rolls.
Coat a very large skillet with 2 teaspoons of the oil. Heat skillet over medium heat for 2 to 3 minutes or until hot. Add 2 sandwiches. Cover sandwiches with foil; weight sandwiches with a large heavy skillet, bacon press, or foil-covered brick.
Cook over medium heat for 8 to 10 minutes or until heated through, turning sandwiches over once and replacing foil and weight. Reduce heat to medium-low, if necessary. Transfer to a serving plate. Keep warm. Repeat with remaining oil and sandwiches. Makes 4 sandwiches.