Hide year for birthdays but not other dates

I am creating an internal/private Employee Directory for my company using this plugin (Circled template), and we would like to incorporate birthdays. E.g., use the related Widget Pack features and list it on their individual profile. However, as most people do not like to disclose their age, we want to exclude the year from showing up in the birthday dates. I could just remove it from the date format in Connections: Settings, but we also want to include (Work) Anniversary dates which do require the year, so it shouldn’t be hidden from ALL dates. Is there a way to target hiding the year portion of a certain type of date and leave the others unaffected?