I know an omelette doesn't technically have to have cheese to be an omelette, but I don't know why you'd want to eat one without one (lactose intolerance excluded). "The more cheese, the better," is my motto and, as such, I'm pretty obsessed with the "inside out parmelet" in the above video. This delicacy from Food Wishes is the slightly indecent love child of frico and eggs, with crispy, browned Parmesan all over the outside. Watch the video for detailed instructions, but basically you shred a bunch of parm and put it in a non-stick pan. Let that start to brown and, just when it starts to get golden, carefully pour beaten eggs over the top. Cover with a lid, and let cook until the eggs are just set. At this point, the video instructs you to go ahead and fold the parmelet in on itself, but I'd probably throw a different, contrasting cheese in there -- maybe some jack cheese, or even American -- for maximum cheesiness.
The Parmelet -- Turning the Late-Night Omelet Inside Out [Food Wishes]