There are so many factors that can lead to brittle burger patties, that I wish Alton Brown would cover an episode on it.
My routine is pretty simple. I buy chuck, grind it myself, form it into patties no taller than 1", and then I throw it on a grill pan.
I don't add anything to the patty. I know some people like to add egg and bread crumbs. Some people believe this glues the burger patty so it doesn't break on you. And then there are purists, who believe that such additions to the patty's infrastructure is blasphemous.
I also read from someone that working salt into the patty, will help fuse the patty during the cooking process, but Bobby Flay says that seasoning should be applied on both sides of the formed patty and not on the inside, or it'll be a meat loaf.
Flay also says the burger patty should be refridgerated for 30 minutes, prior to the cooking process. Some believe the patty should be brought up to room temp.
Some say you should really knead the ground beef, so it's properly melded. Others - including Flay - say you should reduce the kneading, because it'll toughen up the burger... even though it seems like that's the point to avoid crumbling?
Some say the burger patty should be around 3/4" tall. Others say it's too thin. Flay himself advocates 3/4", but if you watch any of his videos, it's pretty clear his burgers are around 1.5". They look like softballs, and he also reminds you to make an indentation in the middle of the patty to neutralize inflation.
My burgers are usually around 3/4"-1", and it's so thin, I don't ever worry about it ballooning on me. It never happens, so I don't even bother with depressing the center of the patty.
Finally, the most widely accepted answer to brittle patties, is to use a meat with more fat content. So far, I've used cheap, supermarket beef that was probably extremely lean, and I've also used Choice-graded chuck ribs with pretty good marbling, and honestly, there wasn't much of a difference in the patty's strength.
So far for me, the burger patty that was freshly ground - regardless of the cut/grade - turned out most tender/fragile. It starts cracking on you when you try to flip the patty. I need to be extremely gentle as I scoot the spatula underneath the patty, hold the top of the patty with my free hand and sofly lay it on the grill pan. I absolutely cannot just effortlessly slide the spatula underneath the patty and flip it like Bobby Flay does with his patties.
I also feel like the fact that I grind my own meat, has a lot to do with it, because I didn't have this problem with pre-ground meat from the store. The meat grinder at the store, churns out these fat, relatively unbreakable noodles of ground beef. My meat grinder just spews out little pieces at a time, and while it's definitely softer and more tender than the store-bought stuff, it also comes with a price... the patty itself is extremely vulnerable.