Sorry if someone posted this and I missed it…
These sandwiches were originally, around 1900-1940, as people have said, merely cheese between slices of bread, not flattened, and either run under the broiler or sauteed in a pan.
Somewhere in the 1940s or so? the sandwich press comes into being and grilled cheese sandwiches are made that way, probably in the name of time saving by the sandwich shop. I didn’t search to see if they were there earlier. Perhaps they were. So, once you see it “grilled” by getting compressed, then that’s what you understand needs to be done.
I squish a little bit. I prefer the density of the sandwich that way, and the carmelization of the bread.
And I use 2% milk american “cheese” slices damnit and it’s delicious.
I think that you need to work a few slices of bacon into that sandwich. Or possibly sausage, but that might make it too thick.
I like the cut of your jib. I was thinking of going all Paula Deen on it and replacing the bread with KrispyKreme doughnuts, but bacon … that’s thikning!
I feel like bacon love
I wonder if Havarti would work. How do I make a pressed sandwich without a press?
Use a frying pan and a spatula.
Use anything heat-proof and put a heavy weight on top of it, like a second frying pan.
Got a brick? Wrap it in 2-3 layers of heavy duty aluminum foil, then heat it in the oven till it’s hotsy-totsy.
How about bacon & cheddar bread?
Add more bacon inside, if you’re that sort of person. If you are that sort of person, send a sample to my house. 
I quite often add a slice of tomato(if garden fresh in the Summer) with bacon and mayo. Hard to beat.
This thread inspired me to make a grilled cheese sandwich for lunch today. Margarine, two slices of Kraft American cheese, pressed down with the spatula, on cheapo wheat bread. Delicious. Can’t beat it.
You should have tried making it without the pressing down. I think you would find it to have been equally delicious. 
One thought for those who find it hard to properly cook a grilled cheese sandwich in a frypan without smushing it: you probably are using too high a temperature. This is browning the bread faster than it heats the inside, preventing the cheese from melting. Recommendation: reduce the heat and take more time in making the sandwich. Yummy goodness results.
If you want to taste the bacon love, grill the sandwich in bacon drippings. You have not lived until you have tasted a grilled cheese sandwich that has been made with bacon drippings as the fat.
Of course, you won’t live very long, but you will have lived happily.