By "as gently as possible", America's Test Kitchen means packing the minced meat as if you're "cradling a newborn baby". Here's what that means.
Every time you touch, grind, move, look at ground meat, it starts to release a protein that's really, really sticky called myosin. ... Basically, when you grind beef, you're damaging the meat fibres — so the more you damage it or touch it or pack it, the more of that sticky protein is going to be formed. And the sticky protein sometimes might not be a bad thing — for instance, [for] something like meatloaf, where you want a bit more cohesion. But for a burger, where you're going to bite into it, you want it to almost just hang together. ...
We kind of bundle the meat into mounds and then very gently pack the meat into patties. By "pack," I really mean it's ... hands-off — it's like you're cradling a newborn baby, almost. You have to be very, very gentle with it. The best part of that is the surface of the burger itself is not completely smooth — it's got all these crags and crevices in it. So when you go to cook it, you're going to have a really nice crust that forms on the sides of the beef.
If you're going to go through all the trouble of mincing your own meat, as you should, follow through with gentle handling to keep the burger tender but crusty on the outside.
Check out the article on NPR for more meat tips, including why you should look for air-chilled chicken instead of water-chilled ones.