No In and Outs in my neck of the woods, so I will try to guess...I AM in Wisconsin, so we know a thing or two bout this stuff. I'm guessing that the bread is toasty and crisp and the cheese is melted through and through and has a velvety creamy texture. Not soggy or greasy. Not rubbery.
Start with cheese shredded, and with cheese, bread and butter at room temp. I bet they give the grill a spritz of H2O while the sandwich is cooking on its first side, and put a cover on...the steam will melt the cheese nice. The type of cheese is important...cheddar can't be beat for taste, but can get rubbery and greasy. Probably need to do a blend with a creamier cheese (Monterrey Jack, Gruyere)...your mileage may vary so you will need to experiment a bit.
Thanks for the tips on how to make an awesome standard grilled cheese . I made one just now and it's the best I've ever made.
In N Out's grilled cheese is a little different- They use a hamburger bun, and as far as I can tell the outside part of the bun never touches the grill at all. It's more like a cheeseburger with no burger, yet the cheese is always perfectly melty. It's just typical american cheese, but it's a sandwich that's close to my heart.