The couple, who told their families they were eloping, met Jan. 29, 2019 in the Java Hutt Cafe inside Carilion Roanoke Memorial Hospital in Roanoke, Va. Dr. Hansen, an oncologist and hematologist, was there for a medical conference. Ms. Moore, who was working as a hospice nurse at the time, was wearing scrubs and getting ready to meet the family of a patient.

They were standing next to each other in line. “I thought she was absolutely gorgeous,” he said. “I thought I did not stand a chance, but I made casual chitchat with the hopes of pulling something out of it. She’s a very strong woman and came across as such. That was both scary and attractive.”

He did manage to give her his business card but figured he’d never hear from her again. Then, that night, she texted him: “Hey, it’s Hailey.”

He remembers thinking: “This is the confident woman from the coffee shop.” He added, “I was doing cartwheels and trying to keep myself calm and collected.” Dr. Hansen comes across as thoughtful, approachable and funny, a formal physician who likes to wear extremely informal socks. Describing his sock collection, he said, “I have flamingoes. I’ve got piñata socks on right now. I have palm tree ones. It’s a talking point with work. It’s something to talk about in the room to break the ice, a less serious conversation.”

He and Ms. Moore soon met for dinner at Table 50, a restaurant in Roanoke where she lived at the time. (He lived in nearby Blacksburg.) They talked about medicine and their previous relationships. Both had been married before and divorced, neither happily.