Prepare the cabbage: Remove a few of the outermost leaves of the cabbage, especially if the cabbage is large or the leaves are falling off or noticeably dirtier. Cut cabbage in half lengthwise. Tie twine around each half at the top and bottom to keep together (see picture). Season the top generously with salt (like you would a steak).
Place a medium to large dutch oven over medium to high heat. Add in butter. Most butter is about halfway melted, place each head face-down (cut-side down). Keep heat at medium high to high and cover pot for 3-4 minutes, until it is tender to touch and gives a bit when you push on it.*If cabbage is still quite hard and not tender, cover for an additional minute or two.
Let cabbage sit, without moving, until the bottom is deeply charred, about four more minutes. If you notice the bottom of the pan is noticeably charring, add a tablespoon of butter or neutral oil.
Lightly season with salt and then flip cabbage over and continue to char the other side. At this time, add in the garlic and ginger. Fry until ginger and garlic is brown, about 3-4 minutes.
Using tongs, remove cabbage heads onto serving platter. Cut off twine before serving.