I've hiked that route. You have to come all the way down the Wildcat Valley Trail to the trailhead on Carter Notch Road, which itself is not plowed in the winter...
We were up in this area yesterday, and met a couple of locals who said that Carter Notch Road would be plowed this winter up to the start of the Hall's Ledge Trail -- which is about 0.2 miles past the trailhead for the Bog Brook Trail, a way of connecting with the southern end of the Wildcat River Trail.
They also mentioned that the road is typically plowed at least as far as the paved section, because of the homes and businesses up to that point. (Mountainside Farm, a year-round bed and breakfast, is located
past the end of the pavement, so the road is most likely plowed at least that far.) The paved section of the road ends about 0.5 miles south of the Bog Brook Trail trailhead/parking area.
(Carter Notch Road is now gated where it turns north at the start of the Hall's Ledge Trail; continuing on this section of the road past the gate for about 1.5 miles leads to an intersection with the Wildcat River Trail.)
So, for folks who might want to ski the Bog Brook, Wild River, or Wildcat River Trails (all marked as X-country ski trails) this winter, it may bear checking out.