NOTE: I said Patsy's is on 118 and Lenox in this vid. It's not, it's on 118 and 1st.
The chopped cheese is a Harlem bodega sandwich made with chopped up burger patties and cheese on a hero roll. I grew up in Morningside Heights and lived in Central Harlem up until around 6 years ago, where I'd get my chop cheese from the bodega on 135th and Lenox (I don't know if it's still around). They're originally from Hajji's Deli on 110th and 1st in Spanish Harlem. My sister who lived in Brooklyn until around the same time had never heard of them. I don't think you find them outside the Harlem and the Bronx much (except now there are some cheffy $15 versions available around the city which... kinda misses the point, I think. This is $4-5 late night drunk food.) The delis are usually run by Yemeni immigrants, who also run most of the fried chicken joints in Harlem that I knew of.
I live in the Bay Area now and we don't get NY-style hero rolls out here, but I was craving one so I made it on our local cheap sandwich bread, Dutch crunch. You can make it on any kind of soft, cheap roll. Anything you'd build a sub or hoagie or grinder or whatever-you-call-that-kinda-sandwich-where-you're-from will work. I also added Hatch chiles to mine. You can use pickled peppers if you want something more similar to what you could actually get in a Harlem bodega.
The real key to this sando is seasoning it with powdered chicken bouillon. At least, that's what the bodegas I went to used to use. Wrapping it before cutting it in half is also a good move, as that wrapping compression is what lets you get bites with a bit of everything, and keeps it from all falling apart. Also being drunk before eating. That helps.
These are usually made on a deli-style flattop griddle, and they're made with cheapo pre-formed burger patties that are chopped up with spatulas and onto a split hero roll (that may or may not be toasted in a press) with American cheese, whatever sandwich toppings you want (raw LTO, grilled onions, pickles, pickled banana peppers, mayo, ketchup, mustard - whatever you want). At home you can make it in a cast iron skillet over medium-high heat. I press my roll between two pre-heated pans to mimic a panini press. You want the bread to be a little crispy on the outside, but not super crusty. It's a loose sandwich so if the bread is too crusty is falls apart.