  We introduce a new type of exclusive processes for a better study of
generalized parton distributions (GPDs), which we refer to as
single-diffractive hard exclusive processes (SDHEPs). We advocate a two-stage
framework for picturing SDHEPs based on the separation of scales, which gives a
clear description both kinematically and dynamically. We examine the
sensitivity of the SDHEP to the parton momentum fraction $x$-dependence of
GPDs, and demonstrate it quantitatively with two specific processes that can be
readily measured at J-PARC or AMBER using a pion beam and at JLab using a
photon beam, respectively. Both processes are capable of providing enhanced
sensitivity to the $x$-dependence, overcoming the problem of shadow GPDs, and
disentangling different types of GPDs with various spin asymmetries.
