If you have an existing house with this style then dense pack cellulose or pour in place open cell foam are your only options.
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.
This is why careful planning is very important.
Interview with the contractor at the end.
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 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.
Step 1 add a vapor barrier.
How to vent a gambrel roof using reflectix.