java.time
The other answer uses troublesome old date-time classes now supplanted by the java.time classes.
The java.time classes use ISO 8601 formats by default when parsing and generating Strings that represent date-time values.
An Instant represents a moment on the timeline in UTC with a resolution of nanoseconds.
Instant instant = Instant.parse( "2015-08-10T21:00:00.090Z" );
Adjust into the time zone through which you want to see this moment.
ZoneId z = ZoneId.of( "America/Montreal" );
ZonedDateTime zdt = instant.atZone( z );
You can interrogate for month and day-of-month.
int month = zdt.getMonthValue();
int dayOfMonth = zdt.getDayOfMonth();