A former graduate student of mine, a Roman Catholic like you—and having, like you, a serious interest in the philosophia perennis and the contemplative paths of the world’s orthodox religions—researched the question you pose with some care. And when I say “researched” I don’t mean only with books and in libraries, but by spending time living in monasteries. He came away convinced that the Carthusians are (your word) the most “mystical”, traditional, and serious of the various Catholic orders.

The problem, however, is that unless one is committed to actually becoming a monk it’s impossible for a Catholic layman to share in this particular world and to benefit from its spiritual practices, as visitors are not permitted in Carthusian houses. My former student was in fact a monastic aspirant at the time (and has since joined another order), and it was only as such that he was allowed to spend time with the Carthusians. Until and unless you can in good conscience say you have a monastic vocation, those doors will be closed to you.

This is quite different from what one finds in Orthodox monasticism, where hospitality toward pilgrims has always been stressed and where spiritual advice is more or less readily given to visitors. It’s not uncommon even for the monks of the Holy Mountain—including those living in the sketes and hermitages, and not only cenobites—to have lay disciples. Withdrawn as these fathers otherwise are from the world, their spiritual radii tend to extend rather broadly and with greater impact into the Orthodox world at large.