Would Burning Gaze work through glass and similar transparent but solid barriers? I know the spell does not grant a gaze attack, a past thread I found on the matter focused on this despite it being clear in the spell description.
The rules say that line of effect is only needed when the spell is being cast at the target area or creature so presumably, for example, you could cast Aggressive Thundercloud through an open window then close it and then direct the cloud's movement. The Burning Gaze spell targets the caster (or preferably the caster's familiar) and grants and attack that that requires the caster (or familiar) to look at the target to cause them to burst into flames, it does not describe heat rays or fire bolts being shot from the eyes only that the thing you look at catches fire.
So my point is that when the attacks are made nothing is described as passing from caster to target and no spell is being cast upon the target. Thus only line of sight and not line of effect should be needed.