Hypothyroidism, most commonly caused by Hashimoto’s thyroiditis, is characterized by insufficient production of thyroid hormones and is typically treated with oral levothyroxine (LT4). However, determining patient-specific LT4 dosage remains challenging due to significant inter-individual variability and the need for repeated clinical adjustments. In this study, a stability-aware, data-driven dosing framework is developed for personalized LT4 therapy. The proposed approach combines a Long Short-Term Memory (LSTM) neural network for predicting free thyroxine (FT4) concentrations with a model predictive control (MPC) strategy for dose optimization. To ensure safe and reliable operation, Input-to-State Stability (ISS) conditions are incorporated within the control framework, ensuring boundedness of the system states and stability of the closed-loop system. The method is evaluated using simulated patients generated with Thyrosim, a validated mathematical simulator of thyroid hormone regulation, under varying biological parameters. Simulation results demonstrate that the proposed framework regulates FT4 concentrations toward patient-specific euthyroid levels during the initial phase of treatment and maintains stable hormone levels throughout the simulation period. Furthermore, consistent performance across a heterogeneous population of simulated patients highlights the robustness of the approach. These findings suggest that the proposed ISS-LSTM-MPC framework provides a promising foundation for automated, personalized LT4 dosing and represents a step toward clinically applicable decision-support systems for thyroid hormone therapy.