I'm pretty sure I read this a long time ago in a Hawthorne collection. It's available here: https://www.gutenberg.org/files/512/512-h/512-h.htm as part of MOSSES FROM AN OLD MANSE AND OTHER STORIES.
Beatrice Rappaccini also appears in Theodora Goss's The Strange Case of the Alchemist's Daughter
and the rest of her Extraordinary Adventures of the Athena Club series. I've only read the first of the series, which I thought was very good. The next one is on my massive reading list at my library, along with several dozen other books, which you guys keep adding to.
There's also a Marvel villain, Monica Rappaccini, who is a biochemist who works for MODOK at AIM, generally going up against the Avengers. Don't know anything about her; just one of the links that came up when I started searching for the story. She isn't the Marvel version of Poison Ivy. All of her biochemical and other weapons are scientific gadgets, not inherent like Ivy's. Per Wikipedia, Monica attended the University of Padua, so that's an obvious call-back to Hawthorne. https://en.wikipedia.org/wiki/Monica_Rappaccini
One could certainly imagine a modernized version of this story, where Dr. Rappaccini genetically engineers his daughter to have poisonous skin, saliva, and/or breath. The title from the podcast must have reminded me of something. Earlier today, (before I listened to the recap podcast), I was thinking about some sort of updated vampire story involving a cyberpunk modified assassin who kills her victims by biting them with her venom-injecting teeth or kissing them with her poisonous saliva, which is just straight-up Poison Ivy, now that I read it.