How do I filter on day & month from a date field?
Maybe you could conjure up something with the Regular expression inside the Date Filter, but that is usually not a nice path. 

I haven't tried it but maybe you can do it with Contextual Filters (Under Advanced settings in the third column in Views). There is a separate Contextual filter for month, and for day: 

node.field_date (month) and node.field_date (day)

Configure them with Default values Current date.