If I'm not mistaken the reason the moon eclipses the sun is because the ratio of distance to size, in regards to the moon and sun, is 1:1. And is it not also true that the reason we only ever see one side of thre moon is due to the fact that the moon is tidally locked to the earth? Now, given the above two are correct, would it then not be possible to have a configuration of moon, star, and exoplanet such that there would be a permanent eclipse on said planet?
The coincidence of moon and sun is not perfect, and due to the varying distances of earth observers to the moon, often the moon is smaller than the sun. https://skycalendar.com/tutorial/angulardiameter.html
The coincidence of their visual size is not necessary for there to be eclipses, the earth is visually far larger than the sun when seen from the moon, and creates eclipses on the moon, for instance. To get eclipses, the planetoids must be equal to or larger in subtended angle than their star.
Tidal lock between planetoids is not critical to your question, but you are inferring from that -- that a planetoid pair rotation could be slowed to the point that it locks to its orbital year. The answer is no -- for the planetoid orbital period to match is stellar orbital period, the two radii would have to be roughly equal. This would require, among other things, for the stellar object to be a neutron star or back hole, so not really any radiation to "shadow".
But also, at a radius where the centroid of the two is at equal distance to the stellar object as the separation between the planetoids, then the actual orbital distance between the planetoids and the stellar object would be 3x. The cg of the closer of the pair would be at 1/2 the radius of the joint cg, and the further would be at 1.5X the radius of the joint cg. A 3x difference in planetoid orbital distance would require them to be orbiting at significantly different velocities to avoid orbital decay, breaking their orbital coupling to each other.