Freeze the ribeye for 30 minutes (optional), trim hard fat, and slice against the grain about ⅛ inch thick.
Heat 1 tablespoon oil and 1 tablespoon butter in a large skillet over medium heat. Cook the onions with a pinch of salt for 8 to 10 minutes until soft and golden.
Add the bell pepper and mushrooms and cook 4 to 5 minutes until tender. Transfer to a plate.
Raise the heat to medium-high and add the remaining oil. Cook the steak in batches in a single layer, seasoning with salt, pepper, and garlic powder, for 2 to 3 minutes until browned.
Return the vegetables to the pan and toss. Reduce heat to low and divide into four portions.
Top each portion with two slices of cheese, cover, and cook about 1 minute until melted.
Lightly toast the rolls, then slide each cheesy portion into a roll and serve hot.