The way to make a proper griddled hamburger:
Start with beef that is between 70/30 and 80/20. You need to use a fatty beef because you are going to be doing a few things that will restrict the amount of juice in the burger and the extra fat will mitigate that.
Season your beef and roll into balls less than 1/4 of a pound. Let the size of your patties be determined by the size of your pan, you need them small enough that once you smash you won't be touching the sides of the pan.
You then want to preheat your pan. You want your pan to be as hot as possible. A commercial griddle can range in heat between 375 F and 650 F, so the temp here can vary quite a bit between places, I find hotter aids crust formation and allows the interior stay closer to medium.
Do not use oil. I can't stress that enough. Your beef will put out plenty and oil will fry the burger instead of allowing a seared crust to develop.
Place your balls of meat on the pan and smash them flat with a spatula. You might need a second spatula to remove the patty from the smashing one, as you lack the space space to heat it like you would on a griddle to prevent sticking. Allow it to cook about two to three minutes, then flip and top with cheese, if desired. Two more minutes and remove.
As an added bonus, you can toast/fry some bread in the beef juices to create a patty melt style burger that will be oozing beefy goodness. You can drop the bread in right after the flip and it will be perfect by the time the patty is done.
You shouldn't need any oil when frying a hamburger, but you definitely need to lower the heat. I fry my hamburgers on Medium-Heat in a stainless skillet. Depending on the size of the patty*, I put the patty into a heated up pan and flip after about 6 minutes, then flip again after another 6 minutes, then again after 2 and then it should be done 2 minutes later. This will make a medium-well burger with a little exterior char, but still juicy and tasty.
*I usually do about a 1/3rd pound patty and I let the meat marinate in a generous amount of Worcestershire sauce before hand.
Best Answer
Most cheeses are going to want to melt and ooze out unless they are very hard and stringy cheeses. A hard mozzarella sliced along the strings may work, but I'd try halloumi above all. You can fry halloumi on a pan and it will brown and form a crust, so it is perfect to stuff in a burger patty.