Recent advancements in Large Language Models (LLMs) have positioned them as powerful tools for clinical decision-making, with rapidly expanding applications in healthcare. However, concerns about bias remain a significant challenge in the clinical implementation of LLMs, particularly regarding gender and ethnicity. This research investigates the evaluation and mitigation of bias in LLMs applied to complex clinical cases, focusing on gender and ethnicity biases. We introduce a novel Counterfactual Patient Variations (CPV) dataset derived from the JAMA Clinical Challenge. Using this dataset, we built a framework for bias evaluation, employing both Multiple Choice Questions (MCQs) and corresponding explanations. We explore prompting with eight LLMs and fine-tuning as debiasing methods. Our findings reveal that addressing social biases in LLMs requires a multidimensional approach as mitigating gender bias can occur while introducing ethnicity biases, and that gender bias in LLM embeddings varies significantly across medical specialities. We demonstrate that evaluating both MCQ response and explanation processes is crucial, as correct responses can be based on biased \textit{reasoning}. We provide a framework for evaluating LLM bias in real-world clinical cases, offer insights into the complex nature of bias in these models, and present strategies for bias mitigation.