This is not a Purim joke. Translations of the Book of Esther say the king sentenced people to be "hanged". But modern translations (including Sefaria) say they were "impaled". They translate "ha-etz" as "the stake", not "the tree". Now, historians believe impalement is far more likely than hanging because that is what the Persians did. So which is it? And if "impaled", did earlier translators err or were simply trying not to be indelicate?
According to Tosafos in Eiruvin 48, Haman and his sons were hung without their heads:
והא דכתיב בתרגום מגילת אסתר פרשנדתא איצטליב על תלת אמין וכן כולם Implied question: It says in the Targum of Megilas Esther (which discusses what occupied the 50 Amos from the top of the scaffold from which Haman and his sons were hung) that Parshandasa was hung in three Amos, and likewise for all of them! התם קטועי ראש הוו Answer: Their heads were cut off.
So our modern day hanging of tying a rope around their necks wouldn't have worked.
See also Me'am Loez quoted here that says, about Haman's son's at least, they were killed before they were hung.
"ואת עשרת בני המן יתלו על העץ" “And let the ten sons of Haman be hanged on the gallows.” (9:13) QUESTION: They were already slain; what was her purpose in having them hung? ANSWER: On the eve of the seventeenth day of Nissan, Mordechai hung Haman on the gallows (Megillah 15a, Rashi). The King ordered that his head should be protected from the birds, and he remained on the gallows for the entire year. To further humiliate the Amalikites and frighten the anti-Semites, on the thirteenth of Adarwhen his sons were slain, she asked that they be hung together. This would magnify the already existing public spectacle and convey a good lesson. (ילקוט מעם לועז)
All in all we don't know if they were skewered onto the pole rotisserie style or hung on the pole crucifixion style, but from these classical Jewish sources it does not seem they had nooses tied around their necks.