Breathable (membrane) means it will let water vapor escape. It is not a protection from the sun. Nor does it insulate form the sun.Thanks Freddy, the dormer itself is a wooden framed dormer. The cladding is cedral click cladding, which is a fibre cement cladding.
The roofers supplied a breathable felt and counter batten to allow airflow. I recall them putting the polystyrene type insulation with foil backing on against the dormer too. The bedrooms, whilst warm, are not stupidly hot, just the eaves, so the heat from the tiles are warming the eaves up.
The tiles are natural slate and have new 25x50 battens 25x50 treated counter battens over a modern high performance breathable felt.
It doesn't seem to be very breathable as the temp hit 50.3c
I am worried what will happen when we hit temperatures in the mid to high 30s and whether that will be safe with such high temperatures.
The eaves has new fluffy insulation laying on it, which they added as the previous insulation was quite pathetic.
I took a photo inside the eaves. There is nothing between the breathable felt and the air in the eaves as seen:
View attachment 380368
You may benefit from some through ventilation i.e. OFV's and vented ridge etc.

