In this work, we propose a phase delay approach to estimate the normal and shear fracture compliances utilizing refracted P- and S-waves from full-waveform sonic (FWS) log data generated by a monopole transmitter in a fluid-filled borehole. We derive analytical plane-wave expressions of fracture-induced P- and S-wave phase time delays for inclined compliant fractures, based on which an inversion scheme to infer fracture compliances is built. A parametric test demonstrates that, for realistic fracture compliance values from ${10}^{-14}$ to ${10}^{-11}$ m/Pa, the proposed method is capable of inferring accurate normal and shear compliance estimates for individual fractures with inclinations up to ${\sim }89^{\circ }$. The applicability of the plane-wave technique to refracted waves in FWS experiments is validated by performing numerical wave propagation simulations for a fluid-filled borehole in a hard-rock environment embedding an inclined fracture. Finally, we apply the method to FWS data acquired in the Bedretto Underground Laboratory for Geosciences and Geoenergies (BULGG) in Switzerland along a borehole intersected by a fracture inclined at 71° in a granitic formation. The inferred normal and shear compliances are $(4.20\pm 0.93) {}\times {}{10}^{-13}$ and $(2.96\pm 1.42) {}\times {}{10}^{-13}$ m/Pa, respectively. These values are realistic given the effective fracture scale sensed by refracted sonic waves and consistent with previously reported results for normal compliance of the same fracture. Both the numerical and field experiments show that the accuracy of shear compliance estimates is much more susceptible to fracture inclination than that of normal compliance, thus, underlining the necessity of taking fracture inclination into account for reliable shear compliance estimates.