In small bowl add steak sauce. (Both amounts). Whisk in mayo til smooth and no longer lumpy, then whisk in horseradish. Taste, adjust horseradish to your own heat preferance, set aside. In small saute pan, saute mushrooms and onion in ONE of the Tbls. of olive oil til crisp-tender or soft. Your own personal preferance. Set aside. (About 5 minutes.)
Get (lightly oiled) cast iron griddle (or panini maker) heating til medium-hot. Lay out 2 pieces of bread. Drizzle 1 Tbl. sauce onto each slice, then spread with back of spoon til evenly coated. Pour rest of sauce into tiny bowl for dipping later. Top one slice with 1 slice of cheese. Top cheese with steak, piling on. As much or little as you like. Top with sauted mushrooms and onions. Top onions and mushrooms with second slice of cheese. Place other slice of bread on top, sauce-side down. Brush top of bread lightly with olive oil. Place olive oil-side down onto heated griddle. Brush new top with a little olive oil. Press lightly, once, with spatula. (Or close lid of panini maker.) Brown bottom. Depending on how hot your pan is, this will take about 30 seconds to 1 minute. Check now and then to prevent burning. You will want this lightly toasted with nice grill marks. When bottom is to likeness, carefully turn over with a spatula and brown other side. (*I lift a corner with my spatula, then carefully hold it up with my fingers, then get under it with my spatula so you don't wreck the bread.) Do the same way when the second side is done. If using a panini maker, just use the alloted time for a sandwich. When done, carefully remove and place on a cutting surface and cut in half diagonally. Serve hot with extra sauce for dipping. (Sauce can be used to dip your sandwich in, or great for dipping french fries, too!)
*Whatever steak you have on hand works just fine. Whether it be cooked at the moment, or leftover. I used leftover prime rib, but any good steak or tenderloin can be used.
This is enough for 2 people if serving with soup and/or salad.