Pat the steaks dry thoroughly, wrap in plastic wrap and freeze for 45 minutes to 1 hour. (Tip: Prepare the rest of the ingredients while the steak is freezing.) Using a sharp chef’s knife, slice the steak against the grain as thinly as possible. (Note: The grain runs in different directions on ribeye. Look for the places where it changes directions, cut the steak into pieces based on that, then slice each piece against the grain.)
Melt ½ tablespoon of the butter with ½ tablespoon of the oil in a large cast iron or other heavy bottom skillet over medium-high heat. Add the mushrooms and onion, season with salt and cook, stirring occasionally, until the mushrooms have released their water and begun to turn golden in spots and the onion is beginning to caramelize, 8 to 10 minutes. Add ½ tablespoon oil and the bell peppers, season with salt and pepper and cook, stirring occasionally, until tender, 1 to 2 minutes. Add the garlic and cook, stirring, until fragrant and all the vegetables are very tender and caramelized, about 1 minute longer. Transfer to a bowl.
Melt the remaining butter with the remaining oil in the skillet. Add the meat, season generously with salt and pepper and cook, stirring, until just cooked through and some of the edges are crisp, 1 to 3 minutes. Remove from the heat and stir the vegetables back into the skillet. Sprinkle the cheese on top. Cover and let stand off the heat until the cheese melts, 1 to 2 minutes. (If the cheese doesn’t melt, you can broil the mixture for 1 to 2 minutes, watching carefully to prevent burning.) Serve hot.