Last week 25-year-old Jessica Rowley became one of about a dozen women nationwide to make a highly unusual career move: she was ordained a Catholic priest. Rowley’s ordination—which took place at Eden Theological Seminary, a progressive institution in Webster Groves, Mo.—is approved by the Ecumenical Catholic Communion, a group of churches that decline to recognize the authority of the pope but see themselves nevertheless as Roman Catholic. This week Rowley—who is also married—begins working full-time as an associate pastor at Saints Clare & Francis, a breakaway parish in Webster Groves. The Roman Catholic Church, not surprisingly, does not recognize Rowley...