Insulating an A-frame/cathedral ceiling on exterior of existing building

A homeowner with an A-frame house in a cold mountain climate is trying to figure out how to insulate the roof, which currently has no insulation at all. The roof is built from thick wood beams with tongue-and-groove wood sheathing on top, and the walls have some fiberglass insulation already. Because of how the stairs and interior layout are built, insulating from inside the house would mean tearing out major parts of the building, so the homeowner is looking at doing the work from outside instead, likely while replacing the aging asphalt shingle roof.

The question centers on which outside approach makes sense: applying rigid foam board directly over the existing wood sheathing with a vapor barrier and no vented gap, or building a raised, vented layer with an air space between new framing and the roof deck, filled with foam board or fiberglass-type batts. The concern in either case is moisture — whether damp air could get trapped between the old wood sheathing and new insulation and cause rot or mold over time, since sealed "unvented" roofs need careful design to avoid that problem.

For the reader, this is a reminder that older, unusual roof shapes like A-frames often cannot be insulated with standard attic methods, and that the choice between a sealed system and a vented one has real consequences for moisture control and long-term durability. Anyone with a similar uninsulated cathedral-style roof, especially one with wood plank sheathing and no attic space, faces the same basic decision before any insulation goes in.