Large language models (LLMs) suffer from a lack of decision-making transparency, limiting their deployment in high-stakes domains such as healthcare. We propose a mechanistic interpretability framework that introduces two novel paradigms: Medical Fine-Tuning with Frozen Attention Layers (FTFA) and Posterior Adaptation Transcoders (PAT). FTFA freezes attention layers while fine-tuning only feed-forward network (FFN) parameters, enabling PAT to efficiently adapt pre-trained transcoders on the same data. This approach achieves over 1000× efficiency improvement compared to training transcoders from scratch. We theoretically justify this methodology and demonstrate its cost-effectiveness for cross-domain transfer. Transcoders are sparse autoencoders that replace MLP layers to provide interpretable feature representations. By substituting MLP layers of both base Gemma2-2b and its medical fine-tuned variant with per-layer transcoders, we enable feature-level attribution analysis. Through systematic pruning and node merging of resulting attribution graphs, we construct human-interpretable decision pathways. Our analysis reveals that LLMs employ two parallel mechanisms for medical diagnosis: pattern matching and multi-hop reasoning, with fine-tuned models demonstrating enhanced correct reasoning patterns. This work provides a practical framework for training transcoders on fine-tuned models at minimal cost, enabling broader application of mechanistic interpretability across domains and potentially guiding model training through transcoder-based analysis.