PARIS — The operation in the private clinic off the Champs-Élysées involved one semicircular cut, 10 dissolving stitches and a discounted fee of $2,900. But for the patient, a 23-year-old French student of Moroccan descent from Montpellier, the 30-minute procedure represented the key to a new life: the illusion of virginity. Like an increasing number of Muslim women in Europe, she had a hymenoplasty, a restoration of her hymen, the vaginal membrane that normally breaks in the first act of intercourse. “In my culture, not to be a virgin is to be dirt,” said the student, perched on a hospital...