@Peter_Kelley,
=TEXT(DATE(YEAR(TODAY()),1,1)+NthDay, "MMM D") should be able to address this. This adds the NthDay to the beginning of current year, converts it back to the format you need (Example "May 2"). Please substitute NthDay with the average day of occurrence.
Please note that the formula will give you different result on a leap year, just that this formula is year specific. if you want to avoid that situation, simply replace the formula with following : =TEXT(DATE(2019,1,1)+121, "MMM D").