Add whole cabbage to a large pot and fill with water. Bring to a boil and cook until soft, about 1-1.5 hours. If water doesn't cover cabbage completely, rotate the cabbage occasionally to cook evenly.
Preheat the gas side of the Cuisinart Twin Oaks Pellet & Gas Grill to high. Once soft, remove cabbage from water and drain. When cool enough to handle, slice each head of cabbage in half and place, cut side up, on a parchment paper-lined baking sheet. Drizzle each half with 2 tablespoons of olive oil, then sprinkle with salt and pepper. Place on the grill and let roast until the outside leaves and the tops are blackened, about 1 hour or so. Remove from the grill and let cool for a few minutes.
When ready to serve, gently break apart each cabbage half, then drizzle with tahini, sprinkle with crumbled goat cheese, then toss nuts and seeds over the top. Serve hot.