Research monasteries are where most of the Bright Way's R&D took place. They emerged during the Golden Age, and most major scientific and technological breakthroughs could be traced back to them. Tailstone, neurogel, fusion power, force projectors, leaseminds, and much more were all invented by ascetics living in research monasteries in an effort to further the Bright Way's goal of finding other sophonts.
This, along with their existing control over the power grid, is where the seeds of Bright Ways corruption from religion to megacorp were planted. You need a lot of money to maintain that kind of research, and the obvious solution was to commercialize the resulting inventions to pay the bills.
In the beginning, the religious character of the monasteries was very apparent. ascetics would spend their days alternately chanting prayers as a community, meditating on the mysteries of Creation (i.e. doing research), and practically applying the insights gained from those meditations. During the Age of Decadence the spiritual component of the monasteries atrophied to the point that they were little more than R&D labs, with the exception of pious enclaves like Hearthside where traditions were maintained.
Research monasteries are much like monasteries on Earth. There's a novitiate where potential new ascetics discern whether they are called to enter into religious life, then vows are taken to forego bearing pups (so as not to distract from their duties), to live in community, and maintain the practice of meditative research. Unlike Earth monasteries, research monasteries aren't separated into male and female orders. Both genders live together in the same community. Other monasteries like those of the Knights of the Sun and orders of healers are gender-locked due to the taboos against male healers and female soldiers.