Understanding how patients perceive their physicians is essential to improving trust, communication, and satisfaction. We present a large language model (LLM)-based pipeline that infers Big Five personality traits and five patient-oriented subjective judgments. The analysis encompasses 4.1 million patient reviews of 226,999 U.S. physicians from an initial pool of one million. We validate the method through multi-model comparison and human expert benchmarking, achieving strong agreement between human and LLM assessments (correlation coefficients 0.72-0.89) and external validity through correlations with patient satisfaction (r = 0.41-0.81, all p<0.001). National-scale analysis reveals systematic patterns: male physicians receive higher ratings across all traits, with largest disparities in clinical competence perceptions; empathy-related traits predominate in pediatrics and psychiatry; and all traits positively predict overall satisfaction. Cluster analysis identifies four distinct physician archetypes, from "Well-Rounded Excellent" (33.8%, uniformly high traits) to "Underperforming" (22.6%, consistently low). These findings demonstrate that automated trait extraction from patient narratives can provide interpretable, validated metrics for understanding physician-patient relationships at scale, with implications for quality measurement, bias detection, and workforce development in healthcare.