MADISON, Wis.—Thor Ringler’s bouncing step, red-framed eyeglasses and flowered-print shirts brighten the colorless hallway at the Madison VA Medical Center where he works. When he bursts into patients’ rooms, he makes sure they feel like individuals in a system that can too often feel impersonal.

Mr. Ringler is at the forefront of the Department of Veterans Affairs’ effort to use what is known as “narrative medicine”—in which a patient is asked to tell their life story, and that story is included in their medical record. The...