To get large amounts of fog/cloud you need large flow of warm moist air to be then be cooled in the target area. Easiest is with preexisting volcanism to supply the energy.
Scenario one: Older yet still active volcano.
A large volcano near a large body of water, on the down wind side. This will ensure valleys will have rivers due to high rain fall. Then the volcano needs to be still active/have and there to be lava near surface. If molten rock is near surface close to one or more rivers near the base of the volcano then there would be a source of warm moist air, along with cooler air in the valley bottom.
This implies basaltic lava and newer igneous terrain which tend to not form large canyon systems. This would scenario would tend to have smaller steeper valleys. But more common then scenario two.
Scenario two: Broad hotspot.
A hotspot such as Yellowstone or Hawaii. A river flows through the region which is heated up and provides warm moist air, upon mixing with the cooler surrounding air, clouds form. this is similar to scenario one but assumes a broader area and more flexible terrain. A hotspot can move under sedimentary rocks where grand canyon like valleys can be carved.
This could allow closest to requested features of terrain but would be least likely to have contentious broadly covering cloud. It would be less continuous due to more opportunities to have broad mixing of air. This would imply shifting locations of cloud/fog.
Scenario Three: Mad scientist secret hideout.
To hide secrets, a nuclear power plant is installed in a valley. A system of piping is installed to release steam throughout the valley thus ensuring permanent thick clouds. The planned nature of this allows for much more consistent and thicker cloud cover.
This would have the highest consistent density of cloud by far thus providing the most shade. However this is an deliberately engineered outcome, which might not fit desired plot.
There are other scenarios that are possible. But to get reliable thick cloud/fog a reliable energy source in close proximity to water is required.