Preheat the oven to 400°F.
In a small bowl, mix the butter and miso until combined. Set aside.
Over high heat, warm ¼ cup of the oil in each of two large cast-iron or other ovenproof sauté pans. When very hot, place half of the cabbage wedges, cut side down, in each pan. Do not overcrowd the pans. Leave the cabbage wedges in the pans without turning until the cut sides are charred and lightly blackened, about 3 minutes. If the oil is absorbed by the cabbage and the pan appears dry, add more oil, 1 tablespoon at a time, until the pan is lightly coated.
When lightly charred, turn the cabbage wedges cut side up and spread the miso butter on the cut sides, dividing it evenly.
Cook for about 3 minutes, or until the color matches the first side. When seared on both sides, place the pans with the cabbage in the preheated oven.
Using a spoon, baste every 5 minutes with melted butter from the bottom of the pan. After 10 minutes, season lightly with kosher salt on both sides. The total roasting time should be about 20 minutes, and the cabbage should be very tender and charred.
Remove the cabbage from the oven, sprinkle with the flaky salt, and squeeze the lime wedges over the top. Serve immediately.
