Interview with the contractor at the end.
Insulating a gambrel roof room.
One of the best ways to protect your mansard roof from the problems of ventilation is to use a vapor barrier on the inside.
To insulate the roof you will need to use a combination of insulation to fully protect the roof.
There is no access into the small sidewall areas and thus must be accessed by drilling holes in the sidewalls in this case the roof and using dense pack of poured foam methods to properly insulate.
If you have an existing house with this style then dense pack cellulose or pour in place open cell foam are your only options.
This should be a must see video for venting a gambrel r.
If you do not want any extra living space in your attic then you can install a low height roof but you will also have a small space to install the insulation and the ventilation system.
This is why careful planning is very important.