According to my understanding, hesychasm, along with asceticism, is the official, traditional, and uniformly accepted spiritual path of the Orthodox church. Is there an analogous, well-defined, widely-accepted spiritual path in the Catholic church?
I may be incorrect, but my understanding is that this practice and mode of life is fully in line with Orthodox liturgical and spiritual practice, but is not considered the only way to practice the Orthodox faith.
In Catholicism, there are many approved "rules" of life that form the basis of a monastic order or spiritual community that is fully integrated into and in full union with the church, a monastery, for example. There are also many various lay devotions and personal apostolates, as well as consecrations and third orders that indicate a serious commitment to a particular way of living out a life of faith. Examples would include consecration to the Sacred Heart of Jesus, installation into wearing the brown scapular, the Third Order of Franciscans, praying the Rosary, Eucharistic Adoration, and the spiritual exercises of Saint Ignatius, among many others.
Because we believe that God made all sorts of people who connect with him in many different ways, we are encouraged to find ways to connect with God that resonate with us. This isn't just my opinion, it's what my bishop told me when I asked him earlier this year.
Some practices that are considered universally helpful, including frequent reading of Scripture, prayer, Eucharist, and Reconciliation.