Clinical AI systems require not just point-in-time evaluation but continuous governance: the ongoing practice of monitoring, evaluating, iterating, and re-evaluating performance throughout deployment. We present an end-to-end framework of governance that integrates rubric validation, live deployment feedback, technical performance monitoring, and cost tracking, with controlled experimentation gating system changes before deployment. Applied to Hyperscribe, an EHR-embedded agent that converts ambient audio into structured chart updates, twenty clinicians authored 1,646 validated rubrics across 823 cases. Seven Hyperscribe versions were evaluated through controlled experiments, with median scores improving from 84% to 95%. Analysis of 107 live feedback entries over three months showed feedback composition shifting from 79% error reports and 14% positive observations to 30% errors and 45% positive observations as engineering interventions resolved failures. Median processing time per audio segment was 8.1 seconds with a 99.6% effective completion rate after retry mechanisms absorbed transient model errors. These results demonstrate that continuous, multi-channel governance of deployed clinical AI is both achievable and effective.