Heat a large cast iron pan over medium heat. Add the oil and heat through. Add the cabbage to the pan with one of the cut sides facing down, working in batches if needed.
Sear the cabbage until golden brown, about 3 minutes. Then, season with salt. Also, flip and continue to sear on the other side, about an additional 3 minutes.
Set the cabbage aside on a plate and return the pan to heat. Then, add the butter and allow to melt, then add the shallots and sauté until tender, about 4 minutes.
Stir in the miso and rice vinegar, then add the sake and ½ cup water to the pan. Heat oven to 375ºF (190ºC).
Nestle the cabbage back into the skillet and move the pan to the oven. Cook until the cabbage has reduced in size and is tender, about 1 hour.
Season with salt and pepper then garnish with sesame seeds and arugula. Serve hot drizzled with the extra sauce.