How do I choose a contemplative lay order? And do I have to be Catholic to do so?

A reader recently wrote to me:

Dear Carl, I grew up as a Protestant, but as I’ve journeyed into Contemplative Christianity over the last while I’ve become more amenable to Roman Catholicism. I have been wondering about lay orders and how to connect with the contemplative tradition through a lay order. I was wondering if you could shed some light on which…