I don't find anything special about Denny's hashbrowns, but I've cooked at a lot of restaurants in Las Vegas when I was young and trust me, casinos do more breakfast than Denny's.
Restaurants generally use peeled & shredded russet potatoes. They spread the potatoes over the grill (you can probably use a griddle) pour on liquid butter fat
(cheaper than buying butter and convenient since it's in liquid form and it comes in a bottle) which drips down through the potatoes. They allow one side of the potatoes to brown and stick together. While that is happening, they put salt & pepper on the potatoes. After one side is sufficiently browned, they flip them with a spatula and allow the other side to brown. Then they scoop them up with the spatula and put them on a plate.
A lot of times in Vegas, they also use a weight on the grill like
THIS because it will put pressure on the potatoes and perhaps squeeze a bit of moisture out of them.