“Smothered” dishes abound in the cooking of the southeastern US, and usually, but not always, involve extended cooking that creates some kind of gravy from the pan juices. I’ve cooked versions of this cabbage with bacon and sour cream for years, and while the dairy suggests roots in northern Europe, it’d feel right at home alongside other traditionally “southern” foods. For the Venetian take on smothered, check out cavolo sofegao.
Cut about a quarter pound of bacon into small pieces and start cooking it in a big skillet with a good splash of extra virgin olive oil. When it’s brown and bit crispy, add a chopped onion. Let it cook for a few minutes while you chop half a head of green cabbage. Toss it in, and continue cooking over medium until the cabbage has completely wilted and begun to brown.
Add at least a half cup of sour cream, creme fraiche, or heavy cream (or up to three times that much, aka a cup and a half; depends on how creamy you like stuff). Cook for another 15 minutes or so.