I've never been to KC but what I understand is burnt ends are the over-cooked, crispy ends of the brisket pieces, maybe just from the deckel.
Here's the Wikipedia article:
From other articles online they're produced by putting the chunks of deckel back in the smoker but you can also just trim the end pieces of that cut.
I think what you may need to ask for around here is 'outside cut' of brisket. Robb Walsh mentioned it in his review of Kozy Kitchen a few years back and you're more likely to find what you're looking for at Black owned places where they cook things to death, as the article says, rather than some place trying to emulate the Central Texas style.
That might be what you're looking for.
Hope somebody more knowledgeable than I offers an opinion, but until then, I'll say this...
"Burnt ends" is not a term you hear much in Texas, if ever. My father, a native Texan, retired in the Ozarks of Central Missouri and often while visiting him, I go up to Kansas City where "burnt ends" is on the menu of all the bbq joints. In addition, I took a KCBS (Kansas City Barbecue Society) class on how to judge BBQ cookoffs. One of the cuts we discussed, sampled and judged was "burnt ends." I've asked several pitmasters up there about it, and have spoken with the central Texas bbq folks and what I was told is that in other regions, the term "burnt ends" refers to the top of the brisket - ie, the point, the deckle, the "high end." Assuming this information is correct, if you want "burnt ends" in Texas, you ask for your beef "wet," or "off the high side."
It's the same thing.
As far as I can tell, the difference in the name comes from the difference in the way the final product is cut and served. Up there, it was my experience that usually they smoke the whole brisket, then remove the deckle and set it aside. They trim up the flat, cool it, and put it into a meat slicer and slice it very thinly. Then they chop the deckle into bite-sized chunks, because you can't really slice it in a meat slicer all that well...it's too tender and fatty...and sell those chopped deckle chunks as burnt ends. In Texas, they just put that whole smoked brisket up on the block and slice it all by hand.
While in the midwest, Kansas City most notably, I asked for my brisket by the pound, unsliced. Figured I'd slice it by hand at home, like I often do Texas brisket. Unfortunately, I found that it had not been smoked enough to be tender without slicing it very thinly in the meat slicer. In Texas, the opposite is true. The meat has been smoked long enough, and properly enough, that it's too tender to put into a meat slicer and try to slice into those thin strips. It would fall apart.
The only difference, that I can tell anyway, between Kansas City-style "burnt ends" and Texas-style "wet brisket" is the manner of cutting. In Kansas City, it's cut into chunks. In Texas, it's sliced. If you prefer the chunks, you can ask the pitmaster to cut it that way.
Hope someone else can offer some more information and insight. I think it's a fascinating topic.