Dowd recounted the awkward fan encounter while appearing on Monday's The Ellen DeGeneres Show.
"One would be at the airport when I was heading to [a] flight - you know, when you go down the escalator," Dowd began. "And a lady jumped straight in front of me - with a pointed finger. And she said, 'I know you, I know you. Who are you?'"
For a little while, the woman, who Dowd noted was riding backward on the escalator, kept trying to figure out who she was, asking her various questions. Dowd kept her conversation to a minimum, only telling the woman it might be best to "turn around" and face forward on the escalator.
"Then, she went like this, 'I know you. I know you. I know who you are,'" Dowd recounted using a serious tone. "You're her, aren't you?"
After Dowd answered in the affirmative, the woman turned around and ran off.
"That's how good you are at that role," DeGeneres commented.
DeGeneres also got Dowd talking about The Handmaid's Tale spinoff - The Testaments, which MGM and Hulu have been developing, and which would feature the Aunt Lydia character. After the talk show host brought up how another show means job security, Aunt Lydia, er, I mean, Dowd, said she would ensure she continues in this role, no matter what.
"That feels fantastic. And the other thing I have, and I'm just going to say it right now, I thought to myself I hope they're going to let me play her," Dowd shared. "The problem is that if they don't, the actress who would have been chosen should watch her back because I would hunt her until she gives it up."
Catch Dowd's full appearance, Monday on Ellen.