How can I render a birthday field to display only Day and month
Go to /admin/config/regional/date-time/formats and select Add format. You will be prompted to enter a format string. This is a php date() formatting string. If you enter "F j" (without the quotes" the date format will be the full month name and the date eg April 11. Refer to the manual for a full list of options.

Then go to /admin/config/regional/date-time and select Add date type. Call your new type 'Birthday' and select the date format you just created from the select list.

Under /admin/config/people/accounts/display select your field and configure the display to use the 'Birthday' format.