Same with a hot pastrami on rye...convinced it can't be done on the kitchen stove.
Meat needs to be really thin, almost shaved, and it needs to be pre-heated. You can't take it out of the fridge and toss it on the griddle. My wife likes Rueben's and I've had to use the microwave to pre-heat/warm up some of the components. I will even lightly toast the rye bread to get it so it's not cold and sucking the heat out of everything else I am trying to cook. Warm up the kraut, warm up the corned beef, lightly toast the bread. I make my own dressing using ketchup, mustard, and pickle relish and I make sure it's at room temperature before final assembly/grilling. BTW, it helps to have an inverter based microwave where you really are dialing down the power level and not just modifying the time the magnetron is actually energized.
There is a good chance that the meat they are using has a higher fat content that what you would buy for yourself.
Cheese steak is a little harder but having the roll warmed up is a good start. Meat should also not be right out of the fridge. I tend to grill the onions and peppers first and then add the meat (which had marinated in Worcester sauce) and add the cheese last. The greasy spoons also tend to tent the "stuffing" to make sure it's all hot.
When we used to go out as a group with relatives I would bow out to go to the rest room and then I would stop on the way back to the table to watch the chef's preparing food. J Alexanders is a southern style grille and they NEVER put meat on the grille cold. It was always partly cooked and held in steam baths so the last 5-10 minutes on the grille was to "finish it off". This was primarily done for chicken and pork, beef was just allowed to rest to near room temperature.
I've found the best way to "grille" chicken is to microwave it on a very low setting (10-20% power) for something like 20-30 minutes. I would turn it at 10 minute intervals and check the internal temp. Once it hit 150 I'd take it out and put it on the grille at a low to medium setting and get the skin crispy and get some nice grille marks.
