Cut cabbage lengthwise through the core into 4 wedges, leaving core intact. Lightly oil cabbage wedges and season generously with salt.

Heat Grill Frying Pan on the grill over medium-high until very hot, 3–5 minutes. Place cabbage wedges cut side down in the Grill Frying Pan and cook until charred but not softened, 3–6 minutes. You may need to do this in two batches, depending on the size of your cabbage. Flip each wedge to the other cut side and cook  until lightly charred, 3–6 minutes more. Remove from heat and let cool to room temperature, 20–30 minutes.

Coarsely grate carrots using the large holes of a box grater. Thinly slice radishes crosswise, then cut into matchsticks. Transfer to a large bowl. Thinly slice scallion and reserve in a separate small bowl.

Make the Scallion-Ranch Dressing: Trim root ends and any withered portions of the green tops of scallions. Cut scallions lengthwise in half, then again lengthwise into quarters. Finely chop scallion and transfer to a medium bowl. Add buttermilk, mayonnaise, sour cream, vinegar, garlic powder, onion powder, oregano, salt, pepper, sugar, and whisk to combine. Refrigerate until assembly.

Once cabbage has cooled, trim the core from each wedge, slice in half lengthwise to create 8 wedges, and slice crosswise as thinly as possible. Add to the large bowl with the carrots and radishes. Season generously with salt and pepper.

Add Scallion-Ranch Dressing and half of lime juice. Use tongs to toss to coat. Taste and season with more salt, pepper, and lime juice as needed. Top with reserved sliced scallion.