I just made this the other day, and I like how it turned out, so I figured I'd post it. I've yet to try Lillian's Southern Style Greens, but this seems like an easy second best:
Greens (I used collard, though something a little less tough might be good)
Bacon grease (1 Tbs. or so - this really adds a lot of flavor, though you could use butter I suppose, it just wouldn't be very good)
Brown Sugar (1 Tbs. or so)
Apple Cider Vinegar
Option: Red pepper flakes
Chop the greens into reasonably sized chunks. Throw into hot bacon grease. I added a shake of red pepper flakes at this point. Cook for a little bit, then add a small amount of brown sugar. Cook a bit longer, then add more apple cider vinegar than you think is necessary - the brown sugar really takes the sting out. Cook until done.
I've been saving bacon grease in a salsa jar, rather than our weird old family standby of aluminum cans - this way, I can seal it, keep it in the fridge, and use it later for cooking.