Here's my method. First, rub a well-seasoned cast iron pan with a light coat of oil and heat until the oil is just starting to smoke (400+ºF/204+ºC). Outside of the pan, break the ground beef up into large meatball size pieces, around 2 inches (5 cm) in diameter. Cook only a couple of these meatballs in the pan at a time. For instance, in my 10 inch cast iron pan, I cooked 2 at a time as shown in the image below.
When you place them in the pan, use your spatula to smash the meat down so that it's about 3/8 inch (1 cm) thick. Let it cook for about 1 minute, then scrape your spatula underneath it, being sure to capture all the browned bits, and flip it over to an adjacent spot on the pan that is still hot. Press down and cook for another minute. The end result should be a patty that is well-browned on the outside while still pink on the inside.
You can then break these patties up into a normal ground beef consistency.
What you absolutely do not want to do is throw all of the ground beef into the pan and break it up. Using too much meat and breaking it up greatly increases the surface area of the meat that comes into contact with the pan, resulting in a massive drop in the temperature of your pan. Constantly stirring the broken up meat would make things even worse. By stirring it up, you're constantly bringing colder parts of the meat into contact with the pan, which will rob even more heat. It's also less likely you'd be able to ever recover the amount of heat necessary to brown the meat without also driving out all the moisture in the meat at the same time, making it overly dry (especially ground round).
The "mini-smash-burger" method outlined above because it minimizes the amount of meat that comes into contact with the pan at one time, and the cast iron pan is always able to recover the minimal heat loss that occurs while cooking each meatball/patty.