My hamburger patties fall apart

There are a couple things you can try. First, consider the fat content of your mince. Fat is what helps bind mince together, so too lean, and you get crumbles. Cold can also help a great deal, and you want to start cooking patties as cold as you can get them, near frozen. If you really want to stick with a lean ground meat, then the other option would be to add in a panade- a mix of stale bread soaked in water (or milk, or other liquid of choice) and squished together to make a paste. That helps bind your meat as well. You can also opt to use panko or cracker crumbs. The only thing about using a binder is that you walk a line between a burger patty and a meatloaf patty, if you add too much bread, it's meatloaf. Make sure your pan is nice and hot before your patty hits the heat, so you hear the sizzle. Then leave it cook in peace- it's a stare fry, so don't keep trying to lift edges to peek at the underside. Watch the edges, because they will brown and let you know it's time to flip. When your patty is nicely crusted brown on the bottom and is easy to slide a spatula beneath it, you do the flip, and flip it like you mean it. No slow and gentle roll over here. It should be in a quick go beneath and over, so there is less time to break apart. You could even support the top with another spatula, to stabilize during the flip.