Prepare the grill for direct cooking over high heat (450° to 550°F) and preheat the grill pan.
In a large bowl combine the onion, peppers, garlic, oil, and oregano. Season with salt and pepper and toss to coat evenly. Spread the vegetables on the grill pan in a single layer. Grill over direct high heat, with the lid closed as much as possible, until they start to brown and are tender, about 6 minutes, stirring occasionally. Transfer to a medium bowl.
Cut the steaks lengthwise into ⅛-inch slices. Place the slices in a medium bowl, add just enough oil to coat them lightly, and season with salt and pepper; toss to coat. Grill the steak slices on the grill pan over direct high heat, with the lid closed as much as possible, until the meat begins to curl and brown, 4 to 6 minutes, turning occasionally. Wearing barbecue mitts, remove the pan from the grill and set it down on a heatproof surface.
Brush the cooking grates clean. Lightly brush the cut sides of the rolls with oil and grill them over direct high heat until toasted, 30 seconds to 1 minute. Gently rub the garlic onto the toasted rolls.
Build the sandwiches with cheese, meat, onions, and peppers. Serve warm.