Clinical decision-making reflects diverse strategies shaped by regional patient populations and institutional protocols. However, most existing medical artificial intelligence (AI) models are trained on highly prevalent data patterns, which reinforces biases and fails to capture the breadth of clinical expertise. Inspired by the recent advances in Mixture of Experts (MoE), we propose a Mixture of Multicenter Experts (MoME) framework to address AI bias in the medical domain without requiring data sharing across institutions. MoME integrates specialized expertise from diverse clinical strategies to enhance model generalizability and adaptability across medical centers. We validate this framework using a multimodal target volume delineation model for prostate cancer radiotherapy. With few-shot training that combines imaging and clinical notes from each center, the model outperformed baselines, particularly in settings with high inter-center variability or limited data availability. Furthermore, MoME enables model customization to local clinical preferences without cross-institutional data exchange, making it especially suitable for resource-constrained settings while promoting broadly generalizable medical AI.