Slow formations, characterized by shear-wave velocities lower than those of the borehole fluid, present significant challenges for shear-wave velocity estimation using monopole acoustic logging, primarily due to the absence of critically refracted shear waves. To address this limitation, a borehole full-waveform inversion framework is proposed in this paper, which employs low-frequency monopole excitation to exploit the sensitivity of Stoneley waves to shear velocity. The elastic wave equation is reformulated in cylindrical coordinates as a recurrent neural network structure within a deep learning framework, allowing automatic differentiation for efficient gradient computation without adjoint-state methods. Numerical experiments reveal that while high-frequency monopole data can accurately recover compressional-wave velocities, they fail to resolve shear-wave velocities due to weak Stoneley energy in the high-frequency data. In contrast, strong low-frequency Stoneley waves enable robust and reliable shear-wave inversion. An inversion workflow is further proposed, in which an initial shear-wave velocity model is derived by applying a velocity ratio to the inverted compressional-wave model and subsequently refined through inversion of low-frequency monopole data. The proposed approach yields high-accuracy shear velocity profiles in the near-wellbore region and remains effective under complex geological conditions, including small-scale anomalies and ultra-slow formations. These results highlight the critical role of Stoneley waves in monopole-based inversion and offer a practical solution for estimating the shear-wave velocities of slow and unconsolidated formations.