Cut the heads of lettuce off their stems, keeping them intact at their base.
Cut in half lengthwise, right through the stem end. Brush away any visible dirt, or run under cool water to get rid of any, and shake loosely to get rid of excess moisture.
Preheat the grill to medium-high and brush the cut sides of the lettuce with oil. Sprinkle with salt.
Grill cut side down for a few minutes, flipping once or twice to get a bit of heat on the leafy side, until the cut side is char-marked and the edges of the leaves are charred as well.
To make the dressing, whisk together the mayonnaise, blue cheese (save half for sprinkling overtop if you like), sour cream, cream and a grind of pepper.
Place the grilled lettuce on a platter and drizzle with the dressing; sprinkle with blue cheese, if you set any aside (or add extra).